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Temperature and dislocation stress field models of the LEC growth of gallium arsenide Schvezov, Carlos Enrique 1986

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TEMPERATURE OF  AND  THE  DISLOCATION STRESS F I E L D  L E C GROWTH OF  MODELS  GALLIUM ARSENIDE  by CARLOS  Lie.  en F i s i c a ,  M.A.Sc.  Universidad  in Metallurgy,  A  ENRIQUE  THESIS  Nacional  de R o s a r i o ,  U n i v e r s i t y of B r i t i s h 1983  SUBMITTED  PARTIAL  IN  REQUIREMENTS FOR  THE  SCHVEZOV  DOCTOR  THE  Argentina,  Columbia,  FULFILLMENT DEGREE  1975  Canada,  OF  OF  PHILOSOPHY  OF  i n THE  F A C U L T Y OF  (Department  We  GRADUATE  of M e t a l l u r g i c a l  accept this thesis to t h e r e q u i r e d  THE  ©  UNIVERSITY  Carlos  OF  STUDIES Engineering)  as conforming standard  BRITISH  COLUMBIA  O c t o b e r 1986 Enrique Schvezov,  1986  S  In  presenting  degree  this  at the  thesis  in  University of  partial  fulfilment  British Columbia,  freely available for reference and study. copying  of  department  this or  publication of  thesis by  for scholarly  his  this thesis  or  her  the  I agree  I further agree  purposes  may  representatives.  It  be is  requirements  for  an  advanced  that the Library shall make it that permission  for extensive  granted  head  by the  understood  that  of  my  copying  or  for financial gain shall not be allowed without my written  permission.  Department The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3  DE-6(3/81)  of  11  ABSTRACT  The  temperature  calculated process.  for The  been  good  stress  compared  agreement.  resolved  dislocation  Using  were  crystal  gradients  and  oxide  radius,  in  the  crystal the  by  distribution four-fold conditions  and and  and  length  density the  on  the  Theses  and  at  a  boron  the  in  end  procedure  symmetry i n the  from  of  the  which  growth  include  that  with the the  in  the  the  and  crystal  depending  on  the  The  boron  strongly  crystal  strongly  dislocation  exhibited growth  growth,  was  The  the  crystal  also  crystal  gas  temperature  determining  After  and  temperature  the  crystal.  and  crystal  thickness,  crystal,  adopted.  crystal.  fields  density  curvature  of  numerical  calculate  vertical  configuration.  of  to  oxide  factors  interface  the  used  range  show  distribution  LEC  determined.  shape,  critical  the  measurements  parameters  taper,  been  temperature  dislocation  results  cross-sections  position  the  surrounding  cooling  eight-fold  of  have  by  element  crystal,  were  interface The  been  growing  on  cone  finite  experimental have  fields  produced  calculated  effects  dislocation  dislocation  influenced  gas  were  density  influenced the  others.  thickness,  dislocation  the  diameter,  solid/liquid  distribution  The  determined.  pressure,  in a  distribution  parameters  distribution  based  fields in  stress  crystal  reported  and  model  environmental  length,  are  stresses,  the  GaAs  stress  density  resulting  analysis. to  The  shear  and  growing  calculations  thermoelastic have  a  fields  two-fold,  and  cooling  Ill  TABLE  OF CONTENTS  Page ABSTRACT TABLE  II  OF CONTENTS  LIST  OF F I G U R E S  LIST  OF T A B L E S  LIST  OF SYMBOLS  I l l XI XXV XXVI  ACKNOWLEDGEMENTS  XXX  CHAPTER 1 INTRODUCTION  CHAPTER THE  1  2  GROWTH  2.1  OF BULK  GaAs  The L i q u i d ( LEC )  2.2  CRYSTALS  4  Encapsulated  Czochralski  Technque 4  Classification  of Defects  i n LEC grown  GaAs  crystal  7  2.3  Dislocations  2.4  Effect of Dislocations Devices  CHAPTER THE  8  on P r o p e r t i e s  o f GaAs 11  3  ORIGIN  CHAPTER  i n LEC-GaAs  OF D I S L O C A T I O N S  IN L E C G a A s  CRYSTALS  12  4  DISLOCATION  DENSITY  AND C R Y S T A L  GROWTH  17  IV  4.1  Stresses  i n the Crystal  Due  to Thermal u  Gradients 4.1.1  Crystal  Dimensions.  4.1.2  Thermal  Conditions  4.2  Alloy  CHAPTER MODELS  18 During Crystal  Growth  t o t h e GaAs  OF D I S L O C A T I O N Stress  5.2  22  GENERATION  IN G a A s  Fields .  Calculation  29 29  of D i s l o c a t i o n  Densities  36  6  OBJECTIVES  CHAPTER  20  5  5. 1  CHAPTER  Addition  17  40  7  FORMULATION  OF THE MODELS  43  7.1  Model  of c r y s t a l  growth  7.1.1  Temperature  7.1.1.1  Governing  7.1.1.2  Finite  Element  7.1.2  Stress  Field  7.1.2.1  Governing  7.1.2.2  Requirements  7.1.2.3  Formulas  7.1.2.4  Linear  7.1.2.5  Quadratic  Element  7.1.3  Von M i s e s  and R e s o l v e d Shear  7.2  Modelling  f o r Cooling  45  Field  45  Equations  45  Equations  48  Model  53  and Element  Equations  f o r the i n t e r p o l a t i o n functions....  f o r Stress  Calculations  Elements  53 57 59 60 61  After  Stresses growth  63 67  V  7.3  Analytical Solutions  7.3.1  A n a l y t i c a l QSS  7.3.2  Analytical solutions  7.3.2.1  Plane  7.3.2.2  Axisymmetric  CHAPTER  Strain  Temperature  Field  f o r the  71  Stress  Field  73  Approximation  73  Solutions  74  8  EVALUATION  OF  THE  MODELS  8.1  Programming  8.1.1  Input  8.1.2  Numerical Solutions  8.2  of  8.3  Comparison of Measurements  RESULTS  AND  of  the A n a l y t i c a l  Model  Predictions  with  Analytical 93  Stress  9  76  93  8.2.2  CHAPTER  Parameters  Field  93  Fields  97 Model  Predictions  with  Temperature 107  Temperature Convergency  Model  for Cooling  -  Programming 115  ANALYSIS  120  9.1  Cone  Angle  9.1.1  Effect  of  Cone  9.1.2  Effect  of  Thermal  9.1.3  Effect  of  the  9.1.4  E f f e c t of Profile Crystal  76  86  Evaluation  Temperature  The and  Input  Parameters  Comparison Solutions  8.4  '  and  8.2.1  9.2  71  121 Angle  Heat  on  Symmetry  Conditions Transfer  Non-linearity  Length  Stress  133 146  Coefficient  i n the  155  Temperature 161 170  VI  9.2.1  Effect  9.3  Crystal  9.3.1  Effect  of Radius  9.4  Growth  Velocity  9.5  Thermal  9.5.1  Radius  9.5.1.1  Boron  oxide  thickness  2 1 . 0 mm  205  9.5.1.2  Boron  oxide  thickness  4 0 . 0 mm  207  9.5.2  Radius  9.5.2.1  Boron  oxide  thickness  21 mm  217  9.5.2.2  Boron  oxide  thickness  40  mm  217  9.5.2.3  Boron  Oxide  thickness  50 mm  221  9.6  Gas  9.7  Curvature of the S o l i d - L i q u i d  9.7.1  Convex  9.7.1.1  Crystal  9.7.1.2  Effect  o f E n c a p s u l a n t T h i c k n e s s and G r a d i e n t  245  9.7.1.3  Effect  of c r y s t a l  251  9.7.2  Concave  CHAPTER  10  RESULTS  FOR  of c r y s t a l  Length  on  Stress  Symmetry  Radius  190 on S t r e s s  Symmetry  202  2 7 . 5 mm  Pressure  205  mm  217  and C o m p o s i t i o n  227 Interface  interface  236  Radius  Interface  AFTER  235 236  Length  COOLING  197 199  Conditions  40.0  175  253  GROWTH  Temperature  257  10.1  Ambient  1000°C  258  10.1.1.  Argon  10.1.1.1  Initial  Gradient  GC  10.1.1.2  Initial  Gradient  GC/2  = 35°C/cm  269  10.1.1.3  Initial  Gradient  GC/4  =  276  258 = 70°C/cm  17.5°C/cm  258  VII  10.1.2.  Boron  10.2  Ambient  10.3  Analysis  CHAPTER  11  SUMMARY  AND  Oxide  278  Temperature  800°C/cm  of the Results  283  for Cooling  283  CONCLUSIONS  290  REFERENCES  APPENDIX EFFECT  306  I  OF  DISLOCATIONS  1.1  Effect  1.2  Dislocations  APPENDIX  ON  GaAs  of D i s l o c a t i o n and F e r m i  AND  DEVICES  320  on P r o p e r t i e s  320  Level  327  II  Integration Temperature  by P a r t s Field  of the Element  Equation  f o r the 330  APPENDIX I I I Evaluation Field  APPENDIX  Quadratic  Elements  f o r the  Temperature 332  IV  Evaluation Elements  APPENDIX  of the Matrix  of S t i f f n e s s  Matrix  Elements "  f o r Linear 338  V Element  Calculations  340  VIII  APPENDIX  VI  Calculation  of Resolved  Shear  Stresses  353  APPENDIX V I I Temperature  APPENDIX  Plane  Solution  362  of the Temperature  Field  367  Analytical  Stresses  372  X  ANALYTICAL X.l  Cooling  IX  Strain  APPENDIX  During  VIII  Analytical  APPENDIX  Field  AS ISYMMETRIC  Axisymmetric  SOLUTIONS  Temperature  FOR  STRESSES  375  Field  375 2  X. 2  Analytical  APPENDIX  XI  COMPUTER  PROGRAMS  Axisymmetric  Solution  for 6 =  - p  ...  385  390  XI. 1  Mesh  XI.2  F i n i t e L i n e a r Element Program f o r the Temperature Calculations during Growth F i n i t e L i n e a r Element Program f o r the S t r e s s Calculations F i n i t e Q u a d r a t i c Element Program f o r the S t r e s s Calculations  412  Program f o r the C a l c u l a t i o n s (010) S e c t i o n s  o f t h e MRSS-CRSS i n 425  Program f o r the C a l c u l a t i o n s Sections  o f MRSS-CRSS  XI.3 XI.4 XI.5  XI.6  Generator  390  394 402  i n (001) 428  IX  XI. 7  APPENDIX XII. 1  XII.2  XII. 3  APPENDIX  Program f o r the Numerical E v a l u a t i o n of Temperatures During Cooling  XIII Program f o r the Numerical E v a l u a t i o n Axisymmetric Temperature Fields  XIII. 2  XIII.3  XIII.4  XIII.5  XIII.6  XIII.7  XIII.8  of  Analytical 437  Program f o r t h e Numerical E v a l u a t i o n o f A n a l y t i c a l A n a l y t i c a l Axisymmetric Stresses f o r Radial Temperature Fields  439  Program f o r the Numerical E v a l u a t i o n o f A n a l y t i c a l Axisymmetric Stresses f o r Axisymmetric Temperature Fields  441  XIII  T E M P E R A T U R E AND OUTPUT (GROWTH) XI11 . 1  432  STRESS  Cone A n g l e C L , 10 mm ; BG, 100 C/cm  PLOTS  FROM  THE COMPUTER  PROGRAMS 444  R, 20 mm : B, ; AG, 50 C/cm  C r y s t a l Length R, 2 7 . 5 mm ; B.21 mm C r y s t a l Radius R, 40 mm ; B,  21 mm  ;  ;  Growth V e l o c i t y R 2 7 . 5 mm ; CL , 55 mm T h e r m a l C o n d i t i o n R, C L , 55 mm ; CA, 30  10 mm  ;  AP, 30 a t m . ; 444  AP, 30 a t m . ;  CA, 3 0 ° ;  ;  CA, 3 0 ° ...  AP, 30 a t m  CA, 3 0 ° ;  521  575  B, 21 mm  617  2 7 . 5 mm ; AP, 30 a t m  624  T h e r m a l C o n d i t i o n s R, 40 mm C L , 80 mm ; CA, 3 0 ° ; AP , 30 a t m  655  Argon Pressure, 2 atmospheres. R, 2 7 . 5 mm ; C L , 55 mm ; CA, 3 0 ° ; Curvature of the S o l i d / L i q u i d CA. 3 0 ° ; AP, 30 atm  B,  21 mm  ....  690  Interface 703  X  APPENDIX XIV TEMPERATURE OUTPUT  AND  STRESS  PLOTS  FROM  THE COMPUTER  (COOLING)  PROGRAMS 761  XIV.1  Ambient  Temperature  800°C,  XIV.2  Ambient  Temperature  1000°C  Argon  761 799  XI  LIST  OF  FIGURES  Fig.  2.1  Schematic  Fig.  3.1  T r a n s m i s s i o n X - r a y t o p o g r a p h y i n GaAs - ( a ) and (b) undopgd ; ( c ) T e - d o p e d , ( 1 1 0 ) a x i a l section  13  Radial dislocation density profiles across wafers obtained from t h e f r o n t , m i d d l e , and t a i l o f a c r y s t a l . T h e r a d i a l p r o f i l e s a r e "W" s h a p e d , a n d t h e a v e r a g e EPD i n c r e a s e s f r o m t h e front to the t a i l  19  Mean d e n s i t y o f " g r o w n - i n " d i s l o c a t i o n s i n G a A s s i n g l e c r y s t a l s ( 2 0 - 2 5 mm i n d i a m e t e r ) g r o w n by t h e C z o c h r a l s k i LEC T e c h n i q u e , a s a f u n c t i o n o f dopant c o n c e n t r a t i o n : (1) Te, (2) Sn, (3) I n , ( 4 ) Zn  23  H y p o t h e t i c a l d i s t r i b u t i o n of the dopant ( I n ) a l o n g t h e p u l l i n g a x i s o f t h e i n g o t . The i n d i u m c o n c e n t r a t i o n o s c i l l a t e s a r o u n d a mean v a l u e which i s p r o p o r t i o n a l t o the d i s t a n c e from t h e top of the ingot  26  (a) and (b) - I s o t h e r m s and s h e a r s t r e s s t o p o g r a p h y i n two g a l l i u m a r s e n i d e s i n g l e c r y s t a l grown under d i f f e r e n t c o n d i t i o n s (c) - D i s t r i b u t i o n of Shear S t r e s s e s T ^ T , and ( d ) - d i s l o c a t i o n s d e n s i t y o v e r t h e c r o s s s e c t i o n s over galium arsenide s i n g l e c r y s t a l s . . .  30  T R S S contours GaAs b o u l e  33  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  4.1  4.2  4.3  5.1  5.2  5.3  o f a LEC p u l l i n g  chamber  f o r the top wafer  5  of a  <001>  D i s t r i b u t i o n of the c a l c u l a t e d d i s l o c a t i o n density i n a gallium arsenide single c r y s t a l b e i n g g r o w n i n t h e < 111 > d i r e c t i o n f o r s l i p systems with Burgers d i s l o c a t i o n vectors perpendicular  34  XII  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  7.1  7.2  8.1  8.2  8.3  8.4  8.5  8.6  Flow c h a r t o f t h e model stress fields  used  to derive  the 44  C r y s t a l c o n f i g u r a t i o n and c o o r d i n a t e used i n t h e m a t h e m a t i c a l model  system 47  Flow c h a r t of the computer program t o c a l c u l a t e the t e m p e r a t u r e f i e l d i n t h e c r y s t a l u s i n g a f i n i t e e l e m e n t method  78  Flow chart of the stress calculations quadratic elements  81  Flow c h a r t program  computer program f o r the using finite linear and  o f t h e mesh  generator  computer 84  Flow c h a r t of the computer a n d p l o t RSS i n a v e r t i c a l  program t o c a l c u l a t e (010) plane  Flow c h a r t of the computer program a n d p l o t RSS i n a ( 0 0 1 ) p l a n e  to  calculate 88  E s t i m a t e d r a d i a t i v e and c o n v e c t i o n h e a t t r a n s f e r c o e f f i c i e n t s f o r GaAs/B 0 ( 1 ) , He ( g ) , N (g) and A ( g ) as a f u n c t i o n o f a m b i e n t t e m p e r a t u r e . The n u m e r i c a l labels are the product of the c a r r i e r concentration X c r y s t a l diameter i n u n i t s o f cm  90  T o t a l h e a t t r a n s f e r c o e f f i c i e n t h , i n B ®3 * a r g o n as a f u n c t i o n o f t h e a m b i e n t t e m p e r a t u r e T (1) T o t a l heat t r a n s f e r c o e f f i c i e n t i n ^ °3' (§) Total heat t r a n s f e r c o e f f i c i e n t i n argon p r e s s u r i z e d a t 30 atm  90  2  Fig.  8.7  87  2  an(  2  2  Fig.  8.8  Temperature dependence of the c r i t i c a l s t r e s s f o r d i s l o c a t i o n g e n e r a t i o n i n GaAs : (1) T e - d o p e d m a t e r i a l , n = 2 X 10*® cm (2) T e - d o p e d m a t e r i a l , n = 7 X 10 cm  XIII  (3) Zn-doped m a t e r i a l , (4) undoped m a t e r i a l . . Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  8.9  8.10  8.11  8.12  8.13  8.14  8.15  8.16  9  X  10  1 8  cm  3  92  A c o m p a r i s o n of f i n i t e calculated temperature 0.3 cm  e l e m e n t and analytical curves for h  A comparison of f i n i t e calculated temperature 0.6 cm"  e l e m e n t and analytical curves for h  94  95  Mesh e m p l o y e d i n t h e c a l c u l a t i o n s o f t h e t e m p e r a t u r e f i e l d s s h o w n i n F i g . 8.9 a n d Number o f n o d e s = 4 5 . Number o f e l e m e n t s = 64  8.10 96  C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r/r for r a d i a l temperature f i e l d s (1) F i n i t e element with averaged element (2) F i n i t e element with nodal temperatures (3) A n a l y t i c a l plane s t r a i n (4) Ana 1 y t i c a 1 - a x i s y m m e t r i c  98  C a l c u l a t e d a z i m u t h a l s t r e s s e s as a f u n c t i o n o f r/r for r a d i a l temperature f i e l d s (1) F i n i t e element with averaged element temperatures (2) Finite element with nodal temperatures (3) Ana 1 y t i c a 1 - p 1 a n e s t r a i n (4) A n a l y t i c a l axisymmetric  99  C a l c u l a t e d a x i a l s t r e s s e s as a f u n c t i o n o f r / r for r a d i a l temperature f i e l d s (1) F i n i t e element w i t h averaged element temperatures (2) F i n i t e element with nodal temperatures (3) Ana 1 y t i c a l - p i a n e s t r a i n (4) A n a l y t i c a l axisymmetric  100  F o u r s t e p s i n t h e mesh r e f i n e m e n t u s e d a n a l y s e the convergency of the f i n i t e stress calculations. ( a ) NN = 1 5 , NE ( b ) NN = 4 5 , NE = 16 ( c ) NN = 4 5 , NE ( d ) NN = 1 5 3 , NE = 256 (b) Q u a d r a t i c ( a ) , ( c ) and (d) l i n e a r e l e m e n t s  102  to element = 16 = 64. elements  C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r/r f o r d i f f e r e n t numbers of n o d e s and s i z e s of element (1) F i n i t e element w i t h a v e r a g e d  XIV  element temperature nodal temperatures. Q = Q u a d r a t i c element  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  8.17  8.18  8.19  8.20  8.21  8.22  8.23  8.24  (2) F i n i t e element with L = Linear element, 103  C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r/r f o r axisymmetric thermal f i e l d s (l)°Finite element with averaged element temperatures (2) F i n i t e element w i t h n o d a l temperatures (3) Ana 1 y t i c a 1 - p 1 a n e strain (4) A n a l y t i c a l - a x i s y m m e t r i c  105  P o s i t i o n of thermocouples i n GaAs c r y s t a l . B = B o r i c o x i d e l a y e r , a r g o n p r e s s u r e 3.04 R e f e r e n c e 218  108  MPa.  Temperatures m e a s u r e d w i t h t h e r m o c o u p l e s 2, 3 and 4 i n Figure 8.18 as a f u n c t i o n of the r e l a t i v e p o s i t i o n of the thermocouples with the i n t e r f a c e . R e f e r e n c e 218  M e a s u r e d and crystal axis  c a l c u l a t e d temperatures at four c r y s t a l lengths  110  along the I l l  M e a s u r e d and c a l c u l a t e d t e m p e r a t u r e s a d j a c e n t t o the o u t s i d e s u r f a c e o f the c r y s t a l a t f o u r c r y s t a l l e n g t h s . The m e a s u r e d a m b i e n t temperature i s a l s o shown  112  M e a s u r e d and c a l c u l a t e d axial temperature g r a d i e n t s along the c r y s t a l axis at four crystal 1engths  113  Flow c h a r t of the computer program f o r the numerical evaluation of the analytical temperature f i e l d s d u r i n g c o o l i n g of the crystal  117  (a) T y p i c a l t e m p e r a t u g e f i e 1 d o b t a i n e d d u r i n g cooling units are 10 C. (b) and ( c ) Von M i s e s s t r e s s c o n t o u r s (MPa) f o r t h e t e m p e r a t u r e f i e l d given in (a). ( b ) NN = 4 5 1 , NE = 8 0 0 . ( c ) NN = 1 1 0 5 , NE = 2048  118  Q  Von M i s e s S t r e s s c o n t o u r s (MPa) i n v e r t i c a l p l a n e s f o r f i v e cone a n g l e s , (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 4 5 ° , ( d ) 54. 7 ° , ( e ) 6 5 ° . Cone s u r f a c e i n (d) c o i n c i d e s w i t h a (111) p l a n e . Crystal r a d i u s , 20 mm ; c r y s t a l l e n g t h , 10 mm ; 2^3 t h i c k n e s s , 10 ram ; B 0^ g r a d i e n t , 100 C/cm ; a r g o n p r e s s u r e , 30 a t m . ; a r g o n g r a d i e n t , 50°C/cm B  Maximum r e s o l v e d s h e a r s t r e s s (MRSS) c o n t o u r s i n MPa, i n v e r t i c a l (010) p l a n e s f o r f i v e cone angles. (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 4 5 ° , (d) 54.7°, ( e ) 65 . C o n e s u r f a c e i n ( d ) c o i n c i d e s w i t h a ( 1 1 1 ) p l a n e . C o n d i t i o n s a r e t h e same a s i n F i g u r e 9.1  T o t a l r e s o l v e d s h e a r s t r e s s (TRSS) c o n t o u r s i n MPa f o r t h e 45 cone a n g l e c r y s t a l . Compare t h e l a r g e s t r e s s l e v e l s o f t h e TRSS w i t h t h e s t r e s s l e v e l s f o r same c r y s t a l s h o w n i n F i g u r e 9 . 1 ( c ) f o r t h e VMS a n d F i g u r e 9 . 2 ( c ) f o r t h e MRSS  C o n t o u r s o f t h e MRSS (MPa) i n e x c e s s o f (a) CRSS ( y i e l d ) ; ( b ) CRSS (MB) ; ( c ) CRSS ( M B T e ) . S h a d e d r e g i o n s i n d i c a t e a r e a s i n w h i c h t h e MRSS i s l e s s t h a n t h e CRSS  S l i p mode d i s t r i b u t i o n i n t h e ( 0 1 0 ) p l a n e f o r t h e 45 cone a n g l e c r y s t a l c o r r e s p o n d i n g t o t h e MRSS d i s t r i b u t i o n s h o w n i n F i g u r e 9.2(c)  Contours i n MPa o f t h e MRSS i n e x c e s s o f ( a ) CRSS ( Y i e l d ) ; ( b ) CRSS (MB) a n d ( c ) CRSS ( M B T e ) . I n m o s t o f t h e c r y s t a l t h e MRSS i s l a r g e r than t h e c r i t i c a l v a l u e s . T h e bump s h a p e d c o n t o u r s ( A ) g i v e complex s t r e s s d i s t r i b u t i o n s i n p e r p e n d i c u l a r cross-sections  MRSS c o n t o u r s (MPa) i n c r o s s - s e c t i o n s perpend i c u l a r t o t h e c r y s t a l a x i s f o r f i v e cone a n g l e s , (a) 7 . 1 ° , (b) 3 0 ° , ( c ) 4 5 ° , (d) 5 4 . 7 ° , (e) 6 5 ° . S e c t i o n ( a - d ) a r e 7.5 mm f r o m t h e i n t e r f a c e a n d s e c t i o n ( e ) i s 8.0 mm f r o m t h e i n t e r f a c e . The h o r i z o n t a l d i r e c t i o n corresponds to the[100] d i r e c t i o n and t h e v e r t i c a l d i r e c t i o n corresponds t o t h e [ 0 1 0 ] d i r e c t i o n  XVI  Fig.  Fig.  Fig.  Fig.  9.8  9.9  9.10  9.11  C o n t o u r s o f t h e MRSS-CRSS ( Y i e l d ) (MPa) f o r t h e 65 cone a n g l e c r y s t a l . MRSS c o n t o u r s a r e shown i n F i g u r e 9 . 7 ( e ) . A t h i g h cone a n g l e s MRSS l e v e l s a r e l a r g e r t h a n CRSS a t t h e c e n t r e and o u t s i d e p a r t o f t h e wafer  142  S t r e s s c o n t o u r s (MPa) i n cross-section 2.5 mm from t h e i n t e r f a c e , (a) and (b) i n a 7.1° cone angle c r y s t a l , ( c ) i n a 45° cone a n g l e crystal, ( a ) MRSS c o n t o u r s , ( b ) a n d ( c ) MRSS-CRSS ( Y i e l d ) . F o r t h e 7.1 cone a n g l e t h e r e i s e i g h t - f o l d symmetry a t t h e c e n t r e i n (a) which i s n o t seen i n ( b ) b e c a u s e s t r e s s l e v e l s a r e l e s s t h a n CRSS. F o r t h e 45 cone a n g l e t h e r e i s f o u r - f o l d symmetry  145  S l i p mode d i s t r i b u t i o n i n a ( 0 0 1 ) p l a n e corresp o n d i n g t o t h e MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 7 ( c ) f o r t h e 45 c o n e a n g l e c r y s t a l a t 7.5 mm from t h e i n t e r f a c e . The e i g h t - f o l d distribution o f t h e mode a t t h e e d g e o f t h e s e c t i o n i s a s s o c i a t e d w i t h t h e e i g h t - f o l d symmetry o f t h e MRSS  147  T e m p e r a t u r e d i s t r i b u t i o n as a f u n c t i o n o f e x t e r n a l temperature g r a d i e n t f o r a 4 5 ° cone angle c r y s t a l . Thermal g r a d i e n t s i n the boron o x i d e a r e : ( a ) 50 C/cm, ( b ) 100 C/cm, ( c ) 200 C/cm a n g ( d ) 400 C/cm. T e m p e r a t u r e s a r e g i v e n i n 10 C. F o r l o w g r a d i e n t s i s o t h e r m s are n e a r l y f l a t and s l i g h t l y convex. F o r l a r g e r g r a d i e n t s i s o t h e r m s a r e c u r v e d and c o n c a v e  149  S t r e s s c o n t o u r s (MPa) i n a ( 0 1 0 ) p l a n e f o r a 4 5 ° cone a n g l e c r y s t a l grown w i t h a 50°C/cm g r a d i e n t i n t h e b o r o n o x i d e , ( a ) MRSS c o n t o u r s d o e s n o t show t h e bump s h a p e d s t r e s s d i s t r i b u t i o n b e l o w t h e s h o u l d e r i n ( a ) . T h e MRSS-CRSS ( Y i e l d ) i s positive o n l y i n a few r e g i o n s i n (b)  151  ( a ) MRSS c o n t o u r s a n d ( b ) MRSS-CRSS ( Y i e l d ) c o n t o u r s f o r a 200 C/cm g r a d i e n t i n t h e b o r o n o x i d e . U n i t s a r e i n MPa. MRSS s t r e s s a r e l a r g e b e l o w t h e s h o u l d e r i n ( a ) . T h e MRSS-CRSS i s g r e a t e r t h a n z e r o i n most o f t h e c r y s t a l i n ( b ) .  152  Q  Fig.  Fig.  Fig.  9.12  9.13  9.14  ( a ) MRSS c o n t o u r s , contours, (c) Slip  ( b ) MRSS-CRSS ( Y i e l d ) mode d i s t r i b u t i o n . G r a d i e n t  XVII  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.15  9.16  9.17  9.18  9.19  9.20  in the boron oxide i s 400°C/cm. In (a) c o n t o u r s are s i m i l a r to those o b t a i n e d f o r a 200°C/cm g r a d i e n t . S t r e s s l e v e l s have d o u b l e d . In (b) only a s m a l l area i n the seed developed stresses l e s s t h a n t h e C R S S . I n ( c ) t h e mode d i s t r i b u t i o n i s s i m i l a r t o t h a t shown i n F i g u r e 9.5 f o r a 100°C/cm g r a d i e n t  154  MRSS c o n t o u r s i n MPa in a cross-section at 7.5 mm f r o m t h e i n t e r f a c e f o r t h e c r y s t a l shown i n F i g u r e 9 . 1 4 ( a ) . The symmetry i s s i m i l a r t o t h a t shown i n F i g u r e 9 . 7 ( c ) f o r a 1 0 0 ° C / c m gradient  156  Temperature p r o f i l e s at t h e s u r f a c e o f 4 5 ° cone angle c r y s t a l s f o r three diferent conditions. Curve H c o r r e s p o n d s to t h e temperatures calcul a t e d u s i n g the o r i g i n a l heat t r a n s f e r c o e f f i c i e n t v a l u e s . I n c u r v e s H / 1.3 a n d H x 1.3 the o r i g i n a l heat t r a n s f e r c o e f f i c i e n t v a l u e s were divided and multiplied by 1.3 respectively  158  Stress contours i n MPa derived from t h e temperature f i e l d o b t a i n e d u s i n g heat transfer c o e f f i c i e n t v a l u e s 1.3 l a r g e r than original values. ( a ) MRSS contours. ( b ) MRSS-CRSS (Yield)  159  S t r e s s c o n t o u r s i n MPa derived from, t h e temperature f i e l d o b t a i n e d u s i n g heat transfer coefficient values 1.3 t i m e s smaller than original values. ( a ) MRSS c o n t o u r s . (b) MRSS-CRSS ( Y i e l d ) . V a r i a t i o n s o f o n l y 20 * a r e o b s e r v e d i n t h e maximum MRSS v a l u e s b e t w e e n ( a ) and F i g u r e 9 . 1 7 ( a )  160  MRSS ( M P a ) c o n t o u r s f o r a c r y s t a l g r o w i n g under four d i f f e r e n t conditions given i n T a b l e 9.3. ( a ) Run #1 ; ( b ) Run #2 ; ( c ) Run #3 ; ( d ) R u n #4. F r o m ( a ) t o ( b ) t h e AMRSS d o u b l e s . I n ( d ) the bump shape below t h e s h o u l d e r does n o t appear  165  MRSS ( M P a ) c o n t o u r s i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 5.0 mm f r o m t h e i n t e r f a c e in t h e c r y s t a l s h o w n i n F i g u r e 9 . 1 9 ( d ) ( r u n #4)  XVI I I  The f o u r - f o l d symmetry symmetry i s o b s e r v e d  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.21  9.22  9.23  9.24  9.25  9.26  9.27  9.28  9.29  with  a slight  two-fold 167  MRSS-CRSS ( Y i e l d ) c o n t o u r s (MPa) f o r t h e f o u r g r o w t h c o n d i t i o n s shown i n T a b l e 9.3 a n d MRSS c o n t o u r s s h o w n i n F i g u r e 9.19 ( a ) Run #1 ; ( b ) R u n #2 ; ( c ) Run #3 ; ( d ) Run #4  169  MRSS (MPa) c o n t o u r s f o r f i v e c r y s t a l lengths. ( a ) 1 3 . 7 5 mm ; ( b ) 2 7 . 5 mm ; ( c ) 5 5 . 0 mm ; ( d ) 8 2 . 5 mm ; ( e ) 1 1 0 . 0 mm . C r y s t a l r a d i u s i s 2 7 . 5 mm  172  MRSS-CRSS ( y i e l d ) (MPa) f o r f i v e c r y s t a l lengths. ( a ) 1 3 . 7 5 mm ; ( b ) 2 7 . 5 mm ; ( c ) 5 5 . 0 mm ; ( d ) 8 2 . 5 mm ; ( e ) 1 1 0 . 0 mm . C r y s t a l r a d i u s i s 2 7 . 5 mm  174  S l i p mode o f t h e MRSS f o r t h e f i v e crystal l e n g t h s s h o w n i n F i g u r e 9.22 ( a ) 1 3 . 7 5 mm ; ( b ) 2 7 . 5 mm ; ( c ) 5 5 . 0 mm ; ( d ) 8 2 . 5 mm ; ( e ) 1 1 0 . 0 mm . C r y s t a l r a d i u s i s 2 7 . 5 mm  176  MRSS-CRSS ( Y i e l d ) (MPa) i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 8.25 mm f r o m t h e c o n e a t f o u r c r y s t a l lengths, ( a ) 1 3 . 7 5 mm ; ( b ) 2 7 . 5 mm ; ( c ) 5 5 . 0 mm ; ( d ) 8 2 . 5 mm. C r y s t a l r a d i u s 2 7 . 5 mm  182  MRSS-CRSS ( Y i e l d ) (MPa) i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 4 9 . 5 mm f r o m t h e c o n e a t two crystal lengths, ( a ) 5 5 . 0 mm ; ( b ) 8 2 . 5 mm. C r y s t a l r a d i u s 2 7 . 5 mm  185  MRSS-CRSS ( Y i e l d ) (MPa) i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 7 7 . 0 mm f r o m t h e c o n e a t two c r y s t a l lengths, ( a ) 8 2 . 5 mm ; ( b ) 1 1 0 . 0 mm. C r y s t a l r a d i u s 2 7 . 5 mm  188  MRSS ( M P a ) c o n t o u r s f o r f o u r c r y s t a l lengths. ( a ) 2 0 . 0 mm ; ( b ) 4 0 . 0 mm ; ( c ) 8 0 . 0 mm ; ( d ) 1 0 0 . 0 0 mm. C r y s t a l R a d i u s 4 0 . 0 mm  193  (a) A x i a l a l o n g t h e c r y s t a l a x i s and (b) r a d i a l t h e r m a l g r a d i e n t s as a f u n c t i o n o f d i s t a n c e  XIX  from t h e i n t e r f a c e f o r ( 1 ) f o r a r a d i u s o f 27 ( 3 ) f o r a r a d i u s o f 40 ( 1 ) 5 5 . 0 mm ; (2 ) 4 0 . 0  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.30  9.31  9.32  9.33  9.34  9.35  9.36  two c r y s t a l r a d i u s . Curve 5 mm ; a n d c u r v e s (2) and 0 mm. C r y s t a l l e n g t h s a r e mm a n d ( 3 ) 8 0 . 0 mm  194  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r f o u r c r y s t a l lengths. ( a ) 2 0 . 0 mm ; ( b ) 4 0 . 0 mm ; ( c ) 8 0 . 0 mm ; ( d ) 1 0 0 . 0 0 mm. C r y s t a l R a d i u s 4 0 . 0 mm  196  Temperature f i e l d s f o r three growth velocities ( a ) 0 . 0 0 0 1 cm/s, ( b ) 0 . 0 0 1 cm/s a n d ( c ) 0.01 cm/s. T e m p e r a t u r e g i v e n i n 10 ° C . L i t t l e change i s o b s e r v e d from (a) t o ( b ) . From ( b ) t o ( c ) g r a d i e n t s h a v e i n c r e a s e d . R a d i u s 2 7 . 5 mm, length 55 mm, e n c a p s u l a n t t h i c k n e s s 21 mm  201  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r t h e t h r e e temperat u r e f i e l d s shown i n F i g u r e 9 . 3 1 ( a - c ) c o r r e s ponding t o three growth v e l o c i t i e s , ( a ) 0.0001 cm/s, ( b ) 0.001 cm/s a n d ( c ) 0.01 c m / s . I n ( a ) and (b) t h e s t r e s s f i e l d s a r e s i m i l a r . In ( c ) the stress distribution and stress values change  204  Temperature p r o f i l e s along the c r y s t a l surface in the environment surrounding the c r y s t a l employed i n t h e c a l c u l a t i o n s . The p r o f i l e s a r e d e r i v e d f r o m F i g u r e 8 . 1 9 , c u r v e 4, r e p r o d u c e d a s c u r v e G. T h e v a l u e s o f g r a d i e n t a r e a v e r a g e d in the crystal length considered. Encapsulant t h i c k n e s s 21 mm  206  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r f o u r temperature p r o f i l e s with average g r a d i e n t s (a) 33°c/cm, (b) 1 7 ° C / c m , ( c ) ll°C/cm and (d) 8°C/cm. R a d i u s 2 7 . 5 mm, l e n g t h 55 mm. Encapsulant t h i c k n e s s 21 mm  209  Temperature p r o f i l e s used i n the c a l c u l a t i o n s f o r two boron oxide thicknesses. ( a ) 40 mm, ( b ) 50 mm. T h e p r o f i l e s a r e d e r i v e d f r o m curve G. T h e v a l u e s o f g r a d i e n t s g i v e n a r e a v e r a g e d along the c r y s t a l length  211  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r t h r e e average gradients. ( a ) 58 C/cm, (b) 29°C/cm, (c)  XX  19 C/cm. R a d i u s oxide thickness  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.37  9.38  9.39  9.40  9.41  9.42  9.43  9.44  2 7 . 5 mm, 40 mm  length  55 mm,  boron 213  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r t h r e e a v e r a g e g r a d i e n t s , (a) 50°c/cm, (b) 25°C/cm, ( c ) 1 7 " C / cm. R a d i u s 2 7 . 5 mm, l e n g t h 55 mm, boron o x i d e t h i c k n e s s 50 mm  215  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r two a v e r a g e gradie n t s , (a) 14°C/cm, ( b ) 7 C/cm. R a d i u s 40 mm, l e n g t h 80 mm, b o r o n o x i d e t h i c k n e s s 21 mm  218  MRSS-CRSS (Yield) (MPa) f o r f o u r a v e r a g e gradients, ( a ) 46 C/cm, ( b ) 2 6 ° C / c m , ( c ) 15°C/cm, ( d ) 9 ° C / c m . R a d i u s 40 mm, l e n g t h 80 b o r o n o x i d e t h i c k n e s s 40 mm  220  mm,  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r a c r y s t a l l e n g t h o f 52 mm c o m p a r a b l e t o t h e b o r o n o x i d e t h i c k n e s s o f 50 mm. R a d i u s 40 mm. A v e r a g e g r a d i e n t 50 C/cm...  222  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r t h r e e average g r a d i e n t s , (a) 43°C/cm, (b) 14°C/cm, ( c ) 7°C/cm. R a d i u s 40 mm, length 80 mm, b o r o n oxide t h i c k n e s s 50 mm  224  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r f o u r temperature p r o f i l e s with average g r a d i e n t s , ( a ) 33 C/cm, (b) 17°C/cm, ( c ) 11 C/cm a n d ( d ) 8 ° C / c m . Radius 27.5 mm, l e n g t h 55 mm. E n c a p s u l a n t thickness 21 mm. A r g o n p r e s s u r e 2 atm  229  Correlation between t h e measured temperature g r a d i e n t s a c r o s s t h e e n c a p s u l a n t and t h e heat t r a n s f e r c o e f f i c i e n t v a l u e s as a f u n c t i o n o f gas p r e s s u r e and gas n a t u r e . Curve A from R e f s . 143144 u s i n g c o n v e c t i o n f r o m a v e r t i c a l w a l l ; and curve B using convection from an horizontal surface  234  Element and n o d a l c o n f i g u r a t i o n a t t h e i n t e r f a c e for c r y s t a l s with curved interface, (a) Convex interface, (b) Concave interface  237  XXI  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.45  9.46  9.47  9.48  9.49  9.50  9.51  9.52  9.53  T e m p e r a t u r e ( 1 0 C) a n d V o n N i s e s s t r e s s ( M P a ) f i e l d s f o r a c r y s t a l with a convex interface shape. Radius 2 7 . 5 mm. Length 2 7 . 5 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. T e m p e r a t u r e profile as s h o w n i n F i g u r e 8 . 1 9 , c u r v e 4  239  MRSS ( M P a ) c o n t o u r s i n t e r f a c e at three 55 mm, ( c ) 8 2 . 5 mm. t h e i n t e r f a c e edge MPa, ( c ) 9.66 MPa. thickness 21mm. P curve 4  240  f o r a c r y s t a l with convex l e n g t h s , ( a ) 2 7 . 5 mm, ( b ) The l a r g e s t s t r e s s e s a t a r e ( a ) 7.22 MPa, ( b ) 9.19 R a d i u s 2 7 . 5 mm, B o r o n o x i d e rofile as i n . F i g u r e 8.19,  MRSS-CRSS ( Y i e l d ) ( M P a ) c o n t o u r s f o r a c r y s t a l with convex i n t e r f a c e a t three l e n g t h s (a) 27.5 mm, ( b ) 55 mm, ( c ) 8 2 . 5 mm. R a d i u s 2 7 . 5 mm, boron oxide t h i c k n e s s 21 mm. P r o f i l e as i n F i g u r e 8.19, c u r v e 4  242  MRSS-CRSS ( Y i e l d ) ( M P a ) c o n t o u r s i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 11 mm f r o m t h e c o n e i n t h e c r y s t a l shown i n F i g u r e 9 . 4 7 ( a )  244  MRSS-CRSS ( Y i e l d ) ( M P a ) c o n t o u r s i n ( 0 0 1 ) p l a n e s at t h r e e d i s t a n c e s form t h e cone. ( a ) 7 9 . 7 5 mm, ( b ) 6 6 . 0 mm, ( c ) 5 2 . 2 5 mm. S e c t i o n c o r r e s p o n d s t o t h e c r y s t a l shown i n F i g u r e 9 . 4 7 ( c )  248  ( a ) MRSS (MP) a n d ( b ) MRSS-CRSS ( Y i e l d ) ( M P a ) f o r a c r y s t a l with convex i n t e r f a c e . Boron oxide t h i c k n e s s 50 mm. R a d i u s 2 7 . 5 mm. L e n g t h 55 mm. Average g r a d i e n t 50°c/cm  249  ( a ) MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) ( M P a ) f o r a c r y s t a l with convex i n t e r f a c e . Boron oxide t h i c k n e s s 50 mm. R a d i u s 2 7 . 5 mm. L e n g t h 55 mm. Average g r a d i e n t 17°C/cm  250  ( a ) MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) ( M P a ) f o r a c r y s t a l with convex i n t e r f a c e . Boron oxide thickness 21 mm. R a d i u s 40 mm. L e n g t h 80 mm. Average g r a d i e n t 55°C/cm  252  (a) for  MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) ( M P a ) a c r y s t a l with convex i n t e r f a c e . (a) Radius  XXI I  2 7 . 5 mm, 21 mm  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  9.54  10.1  10.2  10.3  10.4  10.5  10.6  10.7  10.8  ( b ) 40 mm.  Boron  oxide  thickness 254  MRSS-CRSS ( Y i e l d ) ( M P a ) c o n t o u r s i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 2.75 mm f r o m t h e i n t e r f a c e i n t h e c r y s t a l shown i n F i g u r e 9 . 5 3 ( a )  256  Temperature (10 °C) f i e l d during c o o l i n g at four times. ( a ) 5 s, ( b ) 10 s , ( c ) 20 s , ( d ) 60 s, I n i t i a l g r a d i e n t c l o s e t o i n t e r f a c e 70 C/cm. Argon t e m p e r a t u r e 1000°C  260  MRSS-CRSS ( Y i e l d ) (MPa) d u r i n g c o o l i n g f o r t h e f o u r t e m p e r a t u r e f i e l d s shown i n F i g u r e 1 0 . 1 ( a - d ) . ( a ) 5 s , (b£ 10 s , ( c ) 20 s , ( d ) 60 s . I n i t i a l g r a d i e n t 70 C/cm. A r g o n t e m p e r a t u r e 1 0 0 0 C  263  MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s i n ( 0 0 1 ) p l a n e s at f o u r d i s t a n c e s from t h e bottom ( i n t e r f a c e ) i n the c r y s t a l shown i n Figure 10.2(b) 10 s , ( a ) 2.75 mm, ( b ) 8.25 mm, ( c ) 1 6 . 5 mm, ( d ) 2 7 . 5 mm  268  MRSS-CRSS ( Y i e l d ) (MPa) in (001) p l a n e s at two d i s t a n c e s f r o m t h e b o t t o m i n t h e c r y s t a l s h o w n i n F i g u r e 10.2 ( d ) 60 s , ( a ) 2.75 mm ( b) 8.25 mm  271  3  (a) Temperature f i e l d ( 1 0 ° c ) a n d ( b ) MRSS-CRSS ( Y i e l d ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 1 0 0 0 C a f t e r 10 s . I n i t i a l gradient 35 C/cm  272  (a) Temperature f i e l d ( 1 0 C ) a n d ( b ) MRSS-CRSS ( Y i e l d ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 1 0 0 0 C a f t e r 60 s . Initial gradient 35 C/cm  273  MRSS-CRSS ( Y i e l d ) ( M P a ) i n ( 0 0 1 ) p l a n e s at two d i s t a n c e s f r o m t h e b o t t o m i n t h e c r y s t a l s h o w n i n F i g u r e 10.5 ( b ) 10 s . ( a ) 8.25 mm ( b ) 2 7 . 5 mm  275  3  3 o  (a) Temperature  field  (10  3 o  C)  and  ( b ) MRSS-CRSS  XXI11  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  (Yield ) (MPa^ f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 1 0 0 0 C a f t e r 10 s . Initial gradient 17.5°C/cm  277  MRSS-CRSS ( Y i e l d ) (MPa) i n ( 0 0 1 ) p l a n e s a distance o f 2 7 . 5 mm f r o m t h e b o t t o m i n t h e c r y s t a l shown i n F i g u r e 10.8 ( d ) 10 s  279  (a) Temperature f i e l d ( 1 0 C ) a n d ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i d e a t 1000 C a f t e r 10 s . Initial gradient OoC/cm  280  (a) Temperature f i e l d ( 1 0 ° C ) a n d ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i g e a t 1 0 0 0 ° C a f t e r 10 s . Initial gradient 5oC/cm  281  (a) Temperature f i e l d (10 C) a n d ( b ) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n b o r o n o x i d e a t 1 0 0 0 ° C a f t e r 10 s . Initial gradient 17.5°C/cm  282  10.13 ( a ) T e m p e r a t u r e f i e l d ( 1 0 C ) a n d ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 8 0 0 C a f t e r 10 s . Initial gradient 70 C/cm  284  10.9  10.10  10.11  10.12  3 o  3  3 o  10.14  10.15  (a) Temperature f i e l d ( 1 0 C ) a n d ( b ) MRSS-CRSS ( Y i e l d ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 800 C a f t e r 10 s . Initial gradient 35 C/cm . . 3 o  285  (a) Temperature f i e l d ( 1 0 C ) a n d ( b ) MRSS-CRSS (Yield ) (^Pa) f i e l d f o r a c r y s t a l c o o l i n g i n a r g o n a t 8 OoC a f t e r 10 s . Initial gradient 17.5°C/cm  286  10.16 T e m p e r a t u r e g r o f i l e s i n t h e c r y s t a l c o o l i n g i n a r g o n a t 8 0 0 C f o r two i n i t i a l g r a d i e n t s i n the crystal. (a) 70°C/cm (b) 35°C/cm. The r i g h t p a r t c o r r e s p o n d s t o t h e r a d i a l p r o f i l e s and t h e l e f t p a r t t o t h e a x i a l p r o f i l e . F u l l and b r o k e n l i n e s c o r r e s p o n d t o t h e a x i s and s u r f a c e temperatures respectively  288  3 o  XXIV  Fig.  10.17  Temperature p r o f i l e s i n the c r y s t a l c o o l i n g i n b o r o n o x i d e a t 1000 C f o r two i n i t i a l gradients in the c r y s t a l . (a) 70°C/cm (b) 35°C/cm. The r i g h t part c o r r e s p o n d s to the r a d i a l profiles and t h e l e f t p a r t t o t h e a x i a l p r o f i l e . F u l l and b r o k e n l i n e s c o r r e s p o n d t o t h e a x i s and surface temperatures r e s p e c t i v e l y  289  XXV  LIST  Table  Table  Table  Table  Table  Table  Table  5.1  7.1  7.2  8.1  9.1  9.2  9.3  Resolved Crystals  Shear  Components  Effect Fields  39  i n (001) 65  Values of physical calculations  o f Cone  parameters  in  a  (010) 67  used  i n the 92  Angle  on  Thermal  and  Stress 124  E f f e c t o f A m b i e n t T e m p e r a t u r e on T e m p e r a t u r e G r a d i e n t and S t r e s s  Effect  of N o n - l i n e a r i t y of the  Table  Table  9.5  Effect  Table  VI.1  Rotation t h e RSS  VI.3  Stress  R e s o l v e d Shear S t r e s s Components Plane f o r a [001] C r y s t a l  9.4  Table  and N ) Arseniae  g  Table  VI.2  TABLES  Observed (N ) a n d C a l c u l a t e d (N Dislocation nsities for Gallium Single Crystals  Temperature  Table  OF  Profile  o f Symmetry  of Thermal  performed  150  Ambient  Stress  162  Correlations  Conditions  on  198  Stresses  f o r the c a l c u l a t i o n  226  of 357  Resolved Shear {111} s l i p Compact  on  Axial  form  of  Stress  component  i n the  <110> 358  t h e RSS  from  Table  VI.2  360  XXVI  LIST  (e ) [ B ] *i ' [B']|  Matrix 4 x 2  of d e r i v a t i v e s  Non-dimensional  e )  [C]  Compliance  [C ]  Non-dimensional  1  ~  OF  [B]j  matrix,  SYMBOLS  of the i n t e r p o l a t i o n  e )  matrix,  functions  4 x 2 .  4 x 4 .  [C] m a t r i x ,  4 x 4 .  ( ) e  {d}  Displacement  vector  i n element  e, 2 x  1.  (e) {d}  Nodal  displacement vector  i n element  {d}*  Nodal  displacement vector  a t node  * *red {d } dV  Reduced {d}* vector, 2 x 1 . Global displacement vector i n reduced E l e m e n t a l volume.  E  Young's  e,  2n x  1.  (e )  d  i element  e,  2 x 1  e )  1  Modulus,  form,  2n x  1.  MPa.  (e) {F}  Force  v  1  ( (e e )) {F { F }o.}. i ( ) {F ' } o i F o . l {F '} o v  1  e  ;  vector  f o r element  e,  a e l e m2e nxt 1. e, i t i nno d ee l e mie nitn e, ( ) N o n - d i m e n s i o n a l {F }. vector, 2 x 1 . o 1 (e) E l e m e n t o f {F '} ' vector, o  g  Number  i n element  h  Heat  h  o f nodes  transfer  Convective  force  coefficient  vector,  2n x  1.  (h.t.c.)  h.t.c.  Radiative  h.t.c.  [k]  Stiffness  matrix  ( e) i J [k]..  2 x 2 . Stiffness  submatrix  r  2 x 1 .  e  global  c  1 .  nr i tc ie a lv e cf to or rc e a vte cn to o FI o dr e  Non-dimensional  h  2n x  (e ) f o r element f o r nodes  e, 2n x 2 n . i and j i n e l e m e n t  e,  XXV11  [k']*j*  Non-dimensional  k^j  Element  [K ]  Non-dimensional  (K  Force  1  }  (k}!^  { M ^ *  of  vector  matrix,  matrix. global  stiffness  f o r temperature  S t i f f n e s s m a t r i x from boundary ature f i e l d ca1cua1tions.  [K^]  Stiffness  L  Differential Natural  matrix  matrix.  field  [K ] H  2x2.  calculations.  conditions  for temperature  field  operator.  co-ordinate.  ljj  Length  of  element  at  m  Degree  of  freedom  in  thne  boundary.  element.  ->n  Unit  N  Total  [N]  Interpolation  r r  Radial co-ordinate. Cristal radius.  o  t  normal  vector  number  of  to  nodes  a  surface.  i n the  functions  crystal.  vector.  Time. *  t  non-dimensional  T  Temperature,  T  Q  Reference  time.  °C.  temperature.  T„_ MP  Temperature  Uj  Radial  displacement  of  node i .  u(r,z)  Radial  displacement  at  point  v  Growth  velocity.  -(e)  Volume  of  at  melting  element  point.  r,  e.  Wj  Axial  displacement  of  node i .  w. ( r , z )  Axial  displacement  at  point  Weighting  z.  function.  of  r,  z.  in  temper-  calculations  Axial  co-ordinate.  Coefficient  of  thermal  Non-dimensional  expansion,  °C  non-dimensional  temperature.  Surface  non-dimensional  temperature.  Shear Area  of F o u r i e r  strain  Radial  strain strain  Azimuthal  tensor.  tensor. component.  strain  strain  Eingenvalues Thermal  component.  component. axial  displacement.  of B e s s e l  type  algebraic  equation  diffusivity.  Poisson's  ratio.  Non-dimensional Stress  tensor.  Radial  stress  Azimuthal  Von  i n {£}  equation  (cross-section).  Non-dimensional  Axial  algebraic  tensor.  Initial  Axial  type  component  of element  Strain  .  temperature.  Ambient  Eigenvalues  1  Principal  displacement.  component.  stress  stress  Mises  radial  component.  component.  stress,  MPa.  stresses.  Non-dimentional  radial  stress  Non-dimentional  azimuthal  Non-dimentional  axial  component.  stress  stress  component.  component.  Stress Shear  t e n s o r a t node stress  i , element  e.  component.  Non-dlmentional  shear  stress  component.  XXX  ACKNOWLEDGEMENTS  I  would  Samarasekera the  course  like  for their  of this  Support Universidad acknowledged.  to  form  thank  Dr.  assistance  Fred  Weinberg  and v a l u a b l e  and  Indira  discussions  V.  during  work.  t h e NSERC  Nacional  de  (Canada),  Misiones,  Cominco  Ltd.,  (Argentina)  (Canada)  is  and  gratefully  1  CHAPTER  1  INTRODUCTION  Gallium  Arsenide  is  increasingly  that  semiconductor . . devices  Electron a  become  operating  used  semiconductor  in  the  microwave  consumption  gate with  smaller.  was  which  compound  production  and  of  microelectronic  components  1.1 plus  gates  by  1.6  four  Ten y e a r s  contact  chip  3  may  pads,  materials. than  in  become  I n 1974 t h e GaAs  work  IC'sare  more at  first  I C ' s have  . In a d d i t i o n contain  in  silicon.  circuits  later  and  to the  higher  of that  increases.  o r 4 SKRAM mm  times  tenth  due  both  as t h e VLSI  speed  produced.  ( S i ) are  between  is a  significant  10,000 A  silicon  a r e about  and t h e o p e r a t i n g  produced  active  over  properties  i n GaAs  particularly  GaAs  becoming  GaAs  physical  power  complex  been  in  of  mobilities  with  more  being  crystal  particularly  advantages  differences  These  devices  is a  1-3  The  Si  (GaAs)  than  600  about  10  4 Gbit/sec  . These  facts  indicate  that  GaAs  i s t h e most  promising  5 material In GaAs. of  the  GaAs  generation  optoelectronic  S i does  bright, and  f o r the next  band  n o t glow gap  c a n be lases  devices brightly  structure  made well  of  supercomputers  Si  has  enough  On  of d i f f e r e n t because  the  fewer  applications  and does other  colours  of i t s d i r e c t  hand by  band  not lase GaAs  impurity  than  because  LED s are 1  additions,  gap s t r u c t u r e . 1  At  present  only  about  applications.  By  11  *  1992  of  the  the  GaAs  percentage  chips Is  have  expected  optoelectronic to  Increase  to  7 25  %  . The  GaAs  combination  has  led  to  of micro  the  and  development  optoelectronic of  monolithic  properties  of  optoelectronic  8 -10 integration placed  on  In  of  the  of  15-18 30  %  area  with  an  .  Special  of the multiquantum  of  with  . GaAs  %  presently  circuits  the development  photoconverter at  simple  solar  cells  efficiency  of  solar  cells  monolithic  well  GaAs 20-24  are expected  or  emphasis  hybrid  as  to  an  excellent  compared  reach  tandem  been  laser.  is  %,  has  to S i  efficiencies  cells  which  are  fabricated** .  GaAs  devices  have  advantages  in  radiation  fields  and  at 4  higher times  temperatures. larger  that  temperatures  than  One compared for  Si  12  They  can  Si devices  without  devices,  the  fraction  of  after  total  at present  f o r GaAs  compared  a l lfabricating  difference the  radiation  fields  d e t e r i o r a t i o n , and  o f GaAs  2  ( 5 . 6 cents/mm  ). However  in  at  10  higher  silicon.  of the disadvantages to S i  operate  in  wafer  and  is  cost  costs  i s i t s high t o 0.16  cents/mm  are considered  represents  therefore  cost  only  2  for  a  small  proportionately  less  7 significant  One density,  disadvantage as  Dislocation during  .  o f GaAs,  compared in  many  fabrication  to  cases  and  at present  silicon may  i s i t s high  which  affect  the performance  both  is  dislocation  the  of the  dislocation  yield  devices.  of  free. devices  3  This melt  investigation  grown  during  liquid  stresses  in  to  determine  as  a  to  use  thus  GaAs.  the  this  reduce  of  model the  crystal.  A  thermal  and  the to  on  dislocations  encapsulated  the  function  The  focuses  many  to  Czochralski  growth  (LEC)  mathematical  model  stress  dislocation  and  are  dislocations believed  growth  control  the  fields  in  will the  variables.  The  modify  growth  density  in a  the  present be  generated by  be  growing  objective  controlled  in  thermal developed crystal is  then  conditions  and  way.  4  CHAPTER  THE  2 . 1  The  Liquid  GROWTH OF  2  BULK  GaAs  CRYSTALS  Encapsulated Czochralski  Technque (LEC)  13-15 In 2.1,  a  the  single  the  melt,  the  seed  are  20.8°C.  counter  As  to  "Melbourn"  be  pressure  of  t o grow  raised  to  inside  temperature  atm.  several  of  was  At  50-70  atm.  the  growing  has be  growers,  with  B 0_  crystal.  The  the  inside In  solid  at  at  arsenic  a  much  keep  the  particular  Ltd. is  used  t h e chamber  addition  found  GaAs  therefore  to in  and  melting  formed,  used  i t was  the  Ga  melts  atm. a r e  Instruments  chamber,  Ga The  to maintain stoichiometry.  encapsulating the l i q u i d of  GaAs  required.  2  end  puller.  may  and i n  intermetal 1ic  Pressures  and  purity  temperature  crystal  as  melt  necessary  the  crystals.  Both  high  P r e s s u r e s o f 50  Cambridge  growing  the  this  Once  crystal.  velocities  the  Figure from  atmospheres.  atmospheres  from  GaAs  the  28  Commercial  grower  encapsulation  by  i s 0.98  and  in  i s pulled  a seed  crystal  at  i s 1238°C.  decomposing.  extensively can  pressure  817°C  from  In general  crucible  s y n t h e s i z e GaAs.  pressure from  at  orientation  at d i f f e r e n t  other.  melt  schematically  starting  are rotated  high  GaAs  pressure  required  the  the  shown  controlled  t o each  melts of  ,  upwards,  i n the  in  temperature  GaAs  of  and t h e melt  synthesized  lower  crystal  placed  partial  technique  vertically  directions As  LEC  that  to  the  liquid  the  highest  This  i s done  which  also  covers  the lower  melt  and  encapsulant  o  are  le  Figure  2.1  Schemat i c  of  a LEC  pulling  chamber  contained Using  in a  the high  controlled 8  kgm  orientation  of 10  As single  /  4  an  cm  4  o f As  the  LEC is  permits  the  up t o 75 mm  dislocation  high  1 fi .  puller,  i n diameter densities  pressure  successfully technique  in  grown  crystal at  (LPLEC).  a  in-situ  1-2  In  GaAs  and  i n the  growers,  atmospheres  this  case  which  pressure  crystal  the  weight  growth  oriented  layer  LPLEC  have  the  has  vessel.  the  into  a  produced  starts  o r <111>  i s then  in  growers,  been  process  f o r <100>  The c r y s t a l  synthesis  the B O  With  a n d 14 kgm  melt.  crucible  i n the Melbourn  polycrystal1ine  through  crucible.  usually  with  nitride  A  been novel  17  injection  The  to  separately  '  diameter  crystals  grown  are being the  technique  the  boron  2  material  synthesized  pyrolitic  <100>  alternative  using  starting  a  (HP) LEC p r o c e s s  c a n be  crystals  pressure  or  pressure  i n weight  order  in  quartz  necked  Ga  growers  melt  crystals 18 — 20  with  growth,  LP  a  to reduce  contained of  seed  which  by  100  mm  crystal,  i s dipped  into  the propagation of  21 dislocations slowly  the  the seed  increased  The the  from  growth  diameter power  weighing  to the c r y s t a l  to the f u l l i s monitored  of  the c r y s t a l  input, system  further which  diameter  and t h e d i a m e t e r  of the c r y s t a l .  visually  and w i t h  i s established control  adjusts  then  is  the  done  power  by  a TV  camera.  manual with  control  an  input  with  see  clearly  Once of  automatic computer  2 2-24 control  Human  the  encapsulant,  due  to  capillary  errors,  inabilities  and s y s t e m a t i c forces  25  ,  errors  result  to  i n the weighing  in variations  through procedure  i n the  crystal  7 diameter these  of the order  problems ' 6  When from  '  crystal  the  thermal  1 9  ^ °3  a  n  growth- i s slowly  d  Attempts  are being  is  completed,  cooled  normally  in  the the  crystal  chamber  properties.  to  the material  For  semi-insulating.  additions,  improve  n-type  such  Other  as  is to  solute  additions  n- o r p - t y p e  material  dopants  are  added.  For p-type  of  solute  additions  mechanical  properties  discussed  material and  In  reduce  removed  minimize  some  In, a r e added  to  the  a r e made  to  cases melt the  to melt  semiconductors.  such  a s S, 2  7  Z n a n d Mg  impurities  of LEC-grown  GaAs  S e , T e , S n , S i a n d Ge  a r e added. on  the  crystals  The  effect  electrical  and  and d e v i c e s  are  below.  Classification  A  to  stresses.  i s o e 1 e c t r on i c  2.2  made  .  2 6  z  GaAs  make  o f 5 mm.  defect  structure.  is  Defects  i. i i . i i i . iv.  of Defects  i n LEC grown  defined  as  any  consist  of :  GaAs  deviation  crystal  from  the  Dislocations Chemical  impurities  Deviation Lattice  from defects,  1.  vacancies  2.  subgrains, stacking  stoichiometry  and i n t e r s t i t i a l s grain faults,  ;  boundaries, striations  and o t h e r s  ;  crystal  8 3.  inclusions  and p r e c i p i t a t e s  4.  microdefects, dislocation  loops,  helicoidal These of  The  defects  and  '  .  presented  i n the c r y s t a l ,  multiply.  An  example 3 2 lines  crystallites  o f As  direct  "  3 0  characteristic  lengths  .  of  many  of  Generally which  a  these  defects  .  The  As  i n Ga point  existing precipitates by m o v i n g 2 .3 Dislocations i n LEC-GaAs  diameter  GaAs  of  with  of  i s not  are  matrices  defects  defects  each  small  precipitates  embedded of  number  interact  i s the presence  condensation  Large  and o t h e r s .  31  understood  dislocation  2 8  nature  30 clearly  have  nm  of small  precipitates,  dislocations  100 t o 1  origin  consisting  ;  other  and  precipitates  believed  resulting  or  are  2)  on  to  be  from  sweeping  1) up o f  dislocations.  <100>  single  crystals  produced  by  a 4  Melbourn 2  puller  have  a dislocation  . Considerable or  zero  density  effort large  largest  undoped  15  i n 1985 a c r y s t a l  mm ,  been case VMFEC this  o f a p p r o x i m a t e l y 5x10  3 3  /cm low  density  34  grown  case  i s  fully  a  50mm  and  was g r o w n  includes  diameter  crystal  dislocation  the crystal which  GaAs  i s being  directed  GaAs  grown  crystals.  free  of  i n diameter  striation  using  vertical  encapsulated  free  and  producing  In  1982 t h e  dislocation  was r e p o r t e d '  a modified magnetic  toward  .  In  the  LEC t e c h n i q u e  field.  the melt  was  t o have latter called  The c r y s t a l i s doped.  in The  dislocation with  n,  p  density  or  conditions  crystals  with  diameter  crystal  grown 3  be  reduced  isoe 1e c t r o n i c  thermal  were  can  in  3  pit  7-39  low  .  10°  Impurities.  during  etch  to  growth  densities In  one  vertical  /cm  Recent have  (EPD)  case  by  mm  5x10  <100>  temperature  the  improvements  resulted of  70  doping  3  2  the  undoped for  undoped  gradients  of  in /cm  melt  5  cm  crystals  having  EPD  of  of  the  2  5x10  /cm  over  ingot  length  70  %  of  the  wafer  area  and  throughout  75  %  transverse  sections  4 0 Typical GaAs  dislocation  crystal  are  dislocation  on  dislocations outside entire  etch  pit  in  edge  the  with  crystal  2.2b the  the of  show  the  the  of  and  wafer  33  EPD of  the at the  [110]  is  Figure  along surface  shape  midway  5  cm  the  crystal,  directions  and  maximum  a  the  across (100)  on  degenerates  by  the  shift  to  positions  The  normally  centre  maps  the  wafer  crystal.  symmetry  EPD  symmetry  caused  in  diameter  outside  <110>  between  distribution  of the  4 1  and  showing  distribution a  2.2a  <100>  four-fold  four-fold  distortion  In  minimum  W  end  densities end  [110]  a  has  2.2.  shown  and  shows  tail  on  wafer  giving  the  This  is  outside  higher to  Figure  GaAs  center,  near  symmetry.  minima  (100)  distribution  Close  diameter  in  (EDP)  Figure  taken  wafer.  the  the  wafer.  observed,  fold  a at  and  surface  shown  density  directions  distributions  of  of  the  5.0  cm  to  two-  the  two  closer  to  10  C I 10]  C  1003  b 2.2  a)  Experimentally a l o n g [ 1 1 0 ] and  b)  EPD m a p on a 5cm d i a . w a f e r f r o m t h e t a i l e n d . 1 0 Etch Pits/cm . 1 < y e l l o w < 12 < g r e e n < 15 < b l u e < 20 < b l a c k < 30 . 3 3  determined [010] .  4 1  dislocation  density  4  11  2. 4  Effect  of  Dislocations  Dislocations of  GaAs  in  the  non-radiative  are  on  known  following  Properties  to  affect  ways  recombination  the  1)  GaAs  are  ;  2)  impurity  dislocations active  at  are  the  The  of  the  one  it  has  an  They  of  optoelectronic  been  reported  i t  44  ,  stress,  magnitude  be  absorb  non  Q  defects  ;  structures,  dislocations  lasers  to  3)  When  c a r r i e r s are  not  4 3  how  of  point  cellular  and  applied  order  or  in  of  and  device  lifetime  under  walls  effect  devices, the  arrayed  cell  microelectronic  atoms  properties  considered  A  radiative  Devices  electronic  They  4 2  centers  of  is that  The  properties  depends  fabricated.  In  dislocations the  increase .  the  devices  reduce  4 5 , 4 6  on  for  nature  seriously of  degradation  mechanism  the  optoelectronic  efficiency  the  on  of  LED s 1  of  these  reduce 4 2  ,  and  diodes  by  effects  is  47 unclear With  microelectronic  devices  i t  was  considered  that A  dislocations However believed These  at  did  the  to  results  not  present  affect are  the  significantly time  this  has  electrical  presented  and  affect changed  their and  properties  discussed  in  properties  dislocations of  Q  IC's  Appendix  I.  of  are  GaAs.  12  3  CHAPTER  THE  ORIGIN  Dislocations related.  The  propagation the by  main  with  in  GaAs  are  These  the seed  growth  attributed  related  as  thermal  during  growth.  The  dislocations  stress  near  multiply.  stresses  at  condensation  The  precipitates of vacancies  Examination  of  a  transmission  X-ray  dislocations  are  as  Figures  shown  in  <100>  not  concentrations  Cell  diameters  have  crystal  been  by  eliminated  are  generally  .  As  crystal  of  maximum  along  slip by  and  which  (EPD)  and  that the  center.  In  along  cell in  by the  wafer  previously a W  mm  the  wafer  follows  0.1  thermal by  shows  throughout  diameter  planes  glide.  crystal  described  of  the  regions  concentrating  observed  in  generated  pits  at the o u t s i d e  cells  starting  GaAs°  C  a  into  o r by  loops  3.1 along  form  etch  at  the seed  particles,  grown  a  the  done,  be  distributed  with  dislocations  of  growth  or  propagate  uniformly  density  the  melt  from  from  in  dislocation  topography  dislocation higher  then  foreign  into  result  generated  can a l s o  or  or  i n the c r y s t a l  nucleate  dislocations  related  35  stresses  surface,  CRYSTALS  reduced  i s commonly  dislocations  the  and  c a n be  crystal  to  the c r y s t a l  seed  of d i s l o c a t i o n s  dislocations  seed  GaAs  dislocations  crystal, free  IN L E C  either  related  and m u l t i p l i c a t i o n  a dislocation  The  DISLOCATIONS  seed  crystal.  necking  OF  the  pattern addition walls.  areas  of  13  Figure  3.1  T r a n s m i s s i o n X-ray topography undoped ; (c) Te-doped, (110)  i n GaAs (a^^and axial section  (b)  14  dislocation densities cell  densities  the  wall  cells  of  can  directions  10  5  2  /cm  increase on  a  (based  in  (100)  size  on up  EPD).  to  crystal  3  At  mm  face  lower .  3 0 , 1 1 0  are  The  randomly  oriented.  The  formation  understood. generated climb to  near  after  cells  most the  crystal  their  by  in  constitutional  in  1 1 1  GaAs  wafers  mechanism  walls  and  An  is  in  eventually  energy.  Holmes  the  favoured  solidification  minimize  proposed to  The  of  the  forming  supercooling  at  cell the  not  that  the  glide  cell  structure  and  structure  mechanism  surface,  clearly  dislocations  centre the  alternative  which  is  has  is  been  attributed  similar  to  cell 112  formation because  in  metal  of  the  stoichiometry In arrays at  L.  at  low the  addition of  alloys.  to  interface cell  linear  mechanism  impurity  dislocations  The  This  levels  and  slow  structures are  also  arrays  on  in  GaAs  density  tilt  orientation  the  line.  in  condensation forming  shown  arrays  melt of  as  lines  in  dislocation  of  are  Figure density  loops  in  metal  (10  at 3  /cm ) 2  to  1 1 3  3.1b  having  form on  a  crystal  to  a  small across  lineage  attributed  behind  generally .  linear  Figure  produce  the  similar  observed S,  in  crystals  climb  in  areas  and  0.01°  immediately  which  long  2 /cm  roughly  sometimes 3.1a  about  are  grown  2x10  high  velocities.  shown  observed  than  vacancies  dislocation  Slip  less  difference  These  boundaries  of  GaAs,  crystals,  4 dislocation  unlikely  the  growth  observed  are  appears  sub-  to  the  the  interface,  lineage  boundaries.  etched in  GaAs areas  wafers of  as  lower  15  Specific generation  1.  observations  and  The  and  multiplication  generation  of  in  discussions GaAs  are  of  given  d i s l o c a t i o n s by  dislocation  below  thermal  :  stresses  has  114 been  reported  others  1 1  ^*'  result When  (111)  local  was  to  is  by  probable  dislocations  is  sources  at  the  crystal  Small  precipitates  growing  temperature than  crystal  gradients.  the  critical the This was  12 0  mechanism  stress  the  occurs on 117 118 directions ' . 119 Si by Billig and  for  Penning  in  and  glide  _ <110>  in  co-workers  greater  stress,  proposed  Ge  most  axial  stress  planes  and  stresses  and  shear  mechanism  The  radial  slip  applied  Milvidskii  Thermal  5  from  resolved  2.  * .  1 1  the  by  for  the  concentration surface  of  As  or  in  generation  at the  have  of  heterogeneous bulk  material.  been  found  at  32 dislocations  been  observed  Insular have  Amorphous  on  the  precipitates  cell  p o l y c r y s t a l 1 i n e GaAs  been  observed  associated  with  in  of  Ga  or  As  walls  in  having  diameters  of  GaAs 30  which  Si  dislocation  doped  undoped  generation  .  have 3 0  GaAs 1000°A are  Steinemann  121 and Zimmerly precipitates vacancies  1  1  associate dislocation excess Ge o r i m p u r i t y  of  ^'  1  2  2  in  particular  and  generation with atoms. Excess Ga  native  point  defects  123125 in general which precipitate into dislocation l o o p s h a v e b e e n c o n s i d e r e d as a s o u r c e o f d i s l o c a t i o n s . Lagowski  et  al.  12 6  '  12 7  have  proposed  a  theory  in  which  16  the  Ga  vacancy  formation  of  presented  Is  the  the  in  critical  dislocation  part  b  of  native loops.  Appendix  defect This  in  the  theory  is  I.  o 3.  The  stress  distribution  in  growing  crystals  has  been  12 8 considered used  in  photoelastic  distribution models  to  crystal  and  on  several  local  shear  in  stress  in  X-ray  GaAs  1  2  photo1uminescent In GaAs  summary,  have  melts  been  far  from  formation however  of  several  proposed  dislocation  -  1  supporting  the  chapter.  the  1  causes in  play  densities  3  the  a  for  1 3 4  the  Ga  stress  agreed  correlation has  also  measure  of  1  of been  lattice 3  2  ,  1  3  and  3  normal  dislocations  crystals  vacancy  may  growth role  mechanism.  related  resolved  methods  origin  fundamental glide  to  based  .  l i t e r a t u r e . In the  growing  predicted  The  photoelastic  Under  a  by  ,  a  critical  they  diffraction  stress  mathematical in  dislocations  scanning  dislocations.  evidence next  9  stoichiometry  stresses  to  the  distribution  the  observations.  fields  Indenbom  used  field  distribution  with  using  also  dislocation above  and  observe  stress  levels  well  to  They  the the  The  stress  observed strains  crystals.  derived  reasonably  Nikitenko  methods  establish  stress.  thermal  ways.  in  grown  from  control  the  conditions,  in  controlling  More  experimental  mechanism  is  presented  in  17  CHAPTER  DISLOCATION  The GaAs  density  crystals  parameters. crystal  and is  The  DENSITY  dislocation  quality,  by  1) R e d u c i n g  AND  distribution  strongly  4  CRYSTAL  of  dislocations  dependent density  changing  on  may  be  the growth  the r a d i a l  and  GROWTH  a  in  number  reduced,  conditions  axial  thermal  LEC of  grown growth  improving  the  :  gradients  during  growth 2) A d d i n g  impurities  increasing 3)  Reducing  4)  Maintaining  More may  the  be  to  obtained  i n the c r y s t a l  stoichiometry  dopant by  harden  the  crystal  CRSS  variations  uniform  solution  and  o f t h e Ga  impurity  stopping  radius  fluid  during  a n d As  distribution flow  in  growth  i n the  melt.  i n the  crystal  the  melt  due  to  c o n v e c t i on.  4•1  Stresses  Thermal primarily variables  on  i n the Crystal  stresses 1)  affecting  the  in  Due  the  crystal  to Thermal  crystal  in  dimensions  the temperature  Gradients  LEC  GaAs  and  distribution  2)  growth the  i n the  depend ambient  crystal.  18  4.1.1  Crystal  Dimensions  In  this  category  length,  cone  angle  Increasing the  dislocation  the  case.  diameter  the  15.2  12.7  -  gradients.  to  of  generally  diameters crystals  diameter  in dislocation an  increase  Experimental  of  cm  crystal  large  diameter  diameter,  crystal  dimensions.  the  very  crystal  this  from  did  increases may  5.0  not  -  not  be  7.5  cm  change  the  18  density  due  are  seed  of  At  to  be  and  diameter  the  increase  variables  neck  density.  The  Results  and  Increasing  dislocation  can  the  density in  axial  observations  calculations  are  with  also  increasing  and/or  are  diameter  radial  thermal  contradictory  contradictory.  13 5  13 6  According  to  13 7 Jordan  et  al.  both  axial  and  radial  gradients  increase  with  Buckley-Golden  and  13 5 radius.  On  the  other  hand,  Brice  and  13 8 Humphreys  report  temperature The distance  curvatures  EPD  EPD  values The  is  at  both  effect  the  with  by  end  of  crystal  crystals the  seed cone  Chen  with  and  increasing  a  shape,  throughout  Elliot and  tail  angle  1 1 0  al. of  EPD .  angle  The as  the 113  ends on  Holmes cone  et  up  has  a  Figure In  low  reported  high  crystal. been  found to  in  crystal.  the  with  dislocation  shown  have  They  axial  increases  crystal. W  and  radii.  density,  the  has  gradients  increasing  dislocation  maintained  of  temperature  decrease  seed  density  experimentally decreases  the  across  which  dislocation  axial  indicating  from  distribution 4.1.,  that  that  value  evaluated the of  EPD about  19  Figure  4.1  Radial dislocation density profiles across wafers obtained from the front, middle, and tail of a c r y s t a l . T h e r a d i a l p r o f i l e s a r e "W" shaped, and the a v e r a g e EPD i n c r e a s e s f r o m t h e f r o n t t o t h e tail.  20  25  degrees.  the  EPD  Above  and  cone  Thermal  this  angle.  stresses  Accordingly  the  value This  they agrees  depend  crystal  found  on  with  the  orientation  no  correlation  Watanabe  elastic should  between  39  et a l .  constants  affect  the  of  GaAs.  dislocation 11  density. have  For  found  crystals  that  surfaces  of are  thermal  process  at  the  can  be  oxide and  the  layer  is  ambient  gas  the in  and  pressure  coefficient  depend  in  heat  explain  as  the  transfer the  in 1  1  0  square  environment  having  (100)  geometry  :  and  heat  transfer  heat  transfer  the  are  the  heat  by  around  on  through  The  way  transfer  increases  terms ,  of  which  root  coefficient  variation  taken  directly  crystal. gas  The  pressure,  position  the  the  EPD  3  in  the  of  according  the  between EPD  9  '  1  variation  pressure the  observed.  1  crystal  0  of  to 13 7  increases  be  Growth  simple  the  cannot  for  IC's.  temperature heat  obtained  wafers  crystal.  a  was  et a l .  of  the  system.  explained  transfer  the  thickness,  shielding  of  crystal  of  Plasket  This  crystal  surrounding  characterized  affecting  Increasing effect  the  GaAs  density  Crystal  the  boundary  and  factors  During  in  in  GaAs  production  Conditions  of  direction.  circular for  growth  dislocation  <013>  gradients  coefficients  heater  the  conditions  transfer  boron  in  required  Thermal  Bridgman  lowest  because  Thermal  main  the  grown  advantage  4.1.2  horizontal  R  '  1 1  ^ .  the  This heat  calculations  13 9 ' and  The  change  gas  cannot  21  The reduce  gas  dislocation  temperatures will  pressure  As  result  110,140,141 atomic  inert  He  addition,  changing  used.  As  A  in of  the  nitrogen than  were  found  The  effect  '  is  39  . With  on  at  the  to  high  surface  polycrystalinity  pressure  This  can  changes  the  influence  the  14 2  also  affected gives  other  increase  In  is  efficiencies  gradient  i n the a  in  the  boron  of  the  encapsulant  the  oxide  dislocation  thickness  ,. ,.145-147 gradient .  by  an  EPD  the  type  of  one  order  of  inert  atmospheres  the  order  Kr,  and  Ar,  gases  .. At of  susceptor  cylinder  results  EPD.  The  convection  of  and  this  dislocation  addition the  gas  3°C/cm  15  the  to « 50  B 2  °3  in  of  densities  5  of  change  the  the  /cm  2  through  heat  thermal  axial a  the and  B 0o  crystals  in large  BO  half  the  areas.  in  the  therefore to  reduce  gradients  thickness be  axial  J  crucible radial  the  35,148 used  are  windows  can  layer  Doubling  g r a d i e n t s and  above  for  by  , layers  axial  the  crystal.  reduces  thick  baffles  3  the  layer  diameter 10  different  increasing  mm  lower  reduces  cm  in  30 mm  c  respectively  technique  the  which  grower  density  * present  heating  to  encapsulant.  from  Enhanced  15°C/cm  attributed  c o n v e n t i o n a l Melbourn  decreases  With  material  Ga  and  gas  melt.  atmosphere  argon to  the  reduced  pressure  Excess  twins  be  144  transfer  lower  the  density  higher  counts  143  n  cannot  sufficient  crystal.  surfaces,  dislocation  gas  the  pitted  resistivity  magnitude EPD  leave  of  growth  Without  in  fraction  The  crystal  density.  atoms  j  electrical  during  of  30  produced  to mm.  with  22  4.2  Alloy  The alloy  Addition  t o t h e GaAs  reduction  additions  effectiveness dislocation  or  elimination  to the melt of  was  specific  density  in  of  reported  additions  GaAs  dislocations  (20-25  as  early  in  as  in  reducing  mm  diameter)  GaAs 14 9  1972  the  as  is  by  . The grown  shown  in  effective  in  150 Figure  4.2  reducing  .  It  can  be  seen  that  the d i s l o c a t i o n d e n s i t y  Te  i s the  followed  by  most  I n , Sn  and  Zn.  Seki  151 and  coworkers  effective  determined  in reducing  that  Zn,  S,  Te,  the d i s l o c a t i o n d e n s i t y .  Al  and  N  15 2  15 3  Silicon  are  '  and  154 Boron  have  also  Dislocation N  (20  mm  crystal)  1  5  in 6  ,  been  free  shown  GaAs  diameter) Si  t o have  has been 15 5  (40-50  ,  Te  mm  t h e same  effect.  obtained  doping  (22  in  mm  diameter)  1 5 7  ,  the melt  diameter, In  (50  with <111>  mm  in  ,. . ,35,36,158 diameter)  One n-  or  important  p-type  only  aspect  material.  isoelectronic  promising  impurity  . . , ^. semi-insulating  The  primary  of doping  If semi-insulating  impurities to  decrease  can  effect  of  doping  7  while  40  g/mm  appears  .  f o r Te-doped This  because  ( S i ) GaAs added.  is  produces required,  Indium  density  and  is  the  obtain  . ,159-162 material  f o r d i s l o c a t i o n generation.  16 3  be  the a d d i t i o n  dislocation  stress g/mm  i s that  effect, high  GaAs as  is For  (2x10  explained  temperature  to  increase  undoped  18 by  and  the  GaAs  atoms/cm  3  )  Milvidskii low  critical  the  CRSS  is  the  CRSS  is  et  stresses  a l.  16 3  ,  create  23  favourable  conditions  for  atmosphere  (Cottrell  the  formation  atmospheres)  of  a  stable  impurity  surrounding  moving  C [atom- cnr J 3  Figure  4.2  Mean density of "grown-in" dislocations in GaAs single crystals ( 2 0 - 2 5 mm in diameter) g r o w n by the Czochralski LEC Technique, as a f u n c t i o n of dopant c o n c e n t r a t i o n : (1) T e , ( 2 ) Sn, (3) I n , (4) Zn.  dislocations. effectiveness  This of  the  reduces  an  element  in  2 given atoms  by  a  with  coef f i c i e n t .  factor  U  stopping U  /D  dislocations  where and  dislocation  is D  the is  velocity.  dislocation elastic the  movement  interaction  impurity  The is of  diffusion  24  A  number  account  for  Seki  al.  of  the  specific  effect  of  mechanisms solute  have  atoms  on  been  proposed  dislocation  to  mobility.  151 et  bonds  associated  between  solute  d i s l o c a t i o n movement  elements  with  and  the  matrix,  related  the  bond  breaking with  of  limited  164 success.  Sher  dislocation  et  al.  energies  and  hardness  that d i s l o c a t i o n energies -3 -9 d - d and hardness to used by  to  account  adding  reduce  The  and  reduction  by  not  the  The  cluster  solution In  in  4 >  matrix.  atom is  a  alone  These  remains  ^In As X  a  1 6  GaAs^  calculated  In  ^.  the  by  .  -  be  in and  In  dislocation density predicts  In  GaP  are  in  GaAs  additions  which  is  of  is  21  that  stronger  an %  by  the  .  4  This  effect  solute  than  of  will  by  four  x  in  is  atoms. atoms, a  GaAs  experimental  and  is  tetrahedra is  entity  As  embedded  Ga-As  accounted  five  parameter  as InAs  be  the  sublattice  and  can  cluster  bounded  constant  dilation  a  a  In the  supported  (Ga.In)  roughly  have  atom  by  case  of  configuration  concentration  The  this  consists one  fee  tetrahedra  to  1 6 6  but  results  length  to  length are proportional -11 d This analysis could  dislocation density  tetrahedral  with  X  in  formed  linearly  1  showed to  151155 '  GaN  They  to  unit  reduction  hardening  observations.that  Ga'  the  semiconductors.  (d)  density.  for  InAs  165  BAs  the  for  per -5 d  in  length  increases In-As  bond  varied  in  as  compared  semi q u a n t i t a t i v e l y  solution  hardening  in  metals.  Solute behaviour  of  additions devices  to  the  GaAs  fabricated  on  melt the  can  markedly  crystal  influence  wafer.  the  25  1)  If sufficient could  solute  develop  at  microsegregation  2)  Lineage  crystal  additions  interface  can  be  supercooling  producing  solute  due  to  low  developed  in  the  crystal  variations.  including  at  constitutional  i n the c r y s t a l  orientation  Microdefects, the  the  structure  producing  3)  i s added  the  precipitates solute  levels  can  be  present  in  As  an  example,  Te  present.  produce  stacking  faults  i n the  168-173 crystal  .  prismatic faults  172173 ' .  addition  or n-type  crystals.  well  as  the  that  Si  i s the best  In  this  with  of  of near  no  The  18  the  observed  atoms/cm  10  elements  with  levels  produce  to  stacking  3  same  effect  i n GaAs  ) and  t o GaAs  selected  minimal  with  with  Sn,  dopant 18  f o r GaAs  atoms/cm  3  , as  densities At  a r e made should  other  in dislocation  microdefects.  S  168  small and  Zn  to  produce  compared of  3x10  higher  i n the  produced,  as  I t has  been  n-type  behaviour,  t o Te 3  produce  result  defects  density.  to  /cm  dopant  and  2  S  are  shown  175176  obtained  levels,  above  dislocations  become  3 atoms/cm  pronounced. levels  have  were  high  addition  of elements  dislocations  18 3x10  (2x10  crystal  reduction  case  in  additions  additions  type  levels  Te  at  168,174  desired  at  S  dislocations  of  , .. . . additions  Controlled  additions  dislocations  Helicoidal  p-  Te  ,  straight  Similar  greater  than  and  results 2x10  19  helicoidal  are  obtained  atoms/cm  3  .  with At  In high  dopant In  for  levels  26  (10  20  atoms/cm  3  )  three  different  regimes  of  dislocations  have  17 7 been  reported  the  figure  distance the  by  Pichaud  the  along  In  et  direction  At  4.3  growth  a  end  and  In  along  with  middle  the  shown  (pulling  to axis)  pulling  4.3  . In  increase which  with  changes  axis  of the c r y s t a l ,  periodical  the  the  i n the  associated  and  banded  direction.  oscillations are  seed  with  disappear  i n Figure  Hypothetical d i s t r i b u t i o n of the dopant (In) along the p u l l i n g a x i s o f t h e i n g o t . The i n d i u m concentration oscillates around a mean value which is p r o p o r t i o n a l to the d i s t a n c e from the top of the ingot .  the  tangled  is  shown  density.  Position  Figure  as  concentration  the growth  dislocation  a l.  final  In  levels  convective  distribution  third  dislocation shown flow  d i s l o c a t i o n s are  of  the  density  crystal is  schematically i n the  normal  melt.  to  the  the  bands  reduced.  The  i n the  figure  27  Striation  or  linear  boundaries  are  observed  in  undoped  GaAs  17 8 crystals  .  The  striation  density  can  be  reduced  by  applying  a  17 9 magnetic  field  179180 '  doped field  at  and  thermal  reduced  to  In  0.3°C  not  clear  time  flow  q u a n t i t a t i v e l y change  kilogauss  is  than  this  18 2  simulations  137  doped  oscillations  less i t  kilogauss  and  levels  levels  material.  in 1  8  the  not  not  undoped  With  a  due  to  melt  but  1  the  magnetic  the  crystal 183  1 8 0  .  can  be  However,  and  at  convective Numerical  magnetic flow.  Se  magnetic  characteristic.  convective  ,  convection  field  indicate  suppress  GaAs  vertical  eliminated  how  modelling  do  to  fields  Recent  at  results 184  show The  a  2  KGauss  influence  grown  field  of  crystals  can  stabilize  a  magnetic  is  complex.  field  flow  on  Carbon  the  and  thermal  impurity  fields  distribution  concentrations  are  in  decreased  181 and  Chromium  changed,  as  levels  evidenced  undoped c r y s t a l s The addition  1  without  changing  8  5  of the  Macrosegregation  of  in  In  Figure  with  the  layers  GaAs layer  are  4.3 In  .  .  increased  by  a  In  to  on  the  substrate  MBE  thickness.  exponentially  with  The  which  in  reduce  the  affects  dislocations  deposited  GaAs  critical  increasing  In  presents axis  spacing the  device  resistivity  dislocation  crystal  lattice  in  impurities  electrical  properties  the  surface  misfit  to  along  addition  Specifically, and  GaAs  In  concentration  deposited  reduction  electrical the  Residual  are of  densities  difficulties. will  in  occur  GaAs  quality  of  as  changes  epitaxial  fabrication.  are  above  layer  produced a  critical  thickness  concentration.  For  a  at  the  SI  deposited decreases significant  28  reduction 10  19  the  of  atoms/cm critical  which  dislocations,  3  are  required.  thickness  of  In At  concentrations  concentrations  deposited  material  greater  of  10  is  20  than  atoms/cm  about  1  3  micron  i s small.  The  effect  of  In  alloying  in  FET  characteristic  has  been  18 7 examined alloyed the  by  crystal  wafer.  arrays  Hunter  fabricated  directions  Non are  dislocation  a  FET  post  wafer  or  on  FET's FET's  obtained  solidification velocities  growth.  Bulk  and  uniform  with  of  free by  are  along  process  as  a  gradients  annealing has p r o d u c e d 18 8 10 3 GaAs a n d I n d o p e d GaAs  uniformity  in and  the axial  uniformity be  result  thermal  from  crystals  The  et a l. could  along  expected  radial  doped GaAs 18 8  material  Hunter  annealing low  In  on I n  properties  inhomogeneity  resistivities  dislocation  ion implantation  characteristic  characteristic  properties  direct  showed  wafers  uniform  growth  undoped  . Using  the  Inhomogeneous  resistivity.  the  et a l .  a result of  during  the  with  in of low  crystal  improvements  in  29  CHAPTER  MODELS  Mathematical to  thermal  5.1  for  models  the  as w e l l  Stress  1.  DISLOCATION  GENERATION  for dislocation  stresses  approaches crystal  OF  5  have  been  calculation  IN  GaAs  generation  developed  of  the  as t h e t h e r m o e l a s t i c  glide  using  thermal  stress  by  due  different  field  in  the  field.  Fields  The  temperature  field  in  the  growing  crystal  assuming  the  interface  has  been  18 9 calculated of  numerically  revolution.  experimentally number were and  by  calculated using 5.1  and  RSS  and  (b)  the  right  there  after  plain  a r e shown f o r two  along  strain <111> as a  the  a  dashed  concave the  grown  crystals  crystal.  of  different gives  where  t h e RSS  lines).  The  isotherms  the  isotherm.  are not given.  thermoelastic with  results  The  calculated  i s less  i n parts  Growth In p a r t s  shape  faces  a r e shown  The  i n (a)  number  t h e CRSS  of  and  and  to  to  that  (shown  figure  change  conditions ( c ) and  the  in  isotherms  It i s apparent than  a  stresses  i n the c r y s t a l  t h e CRSS.  at  flat  The  dimensions.  interface  determined  temperatures  crystal  of p o s i t i o n  areas  1200°C  The  paraboloid  were  surface  approximation.  function  following  characteristics  near  cylindrical  of the temperatures few  conditions  the c r y s t a l .  assuming  for a  are a  the  first  a  boundary  measuring  of p o s i t i o n s  Figure  by  The  is a  are  convex crystal  (d) of the f i g u r e  30  0-V  (I  0.1/  0.8  rIK  fi• g  9, 710  Fig.  5.1  (a) and (b) - I s o t h e r m s and s h e a r s t r e s s topography in two gallium arsenide single crystal grown under d i f f e r e n t c o n d i t i o n s . The f i g u r e s n e x t t o t h e c u r v e s a r e x v a l u e s (kg/mm ) c o r r e s p o n d i n g t o t h e constant-stress l i n e s . The d a s h e d c u r v e s outline regions in which the effective stresses are lower than the reduced yield s t r e s s . The f i g u r e s i n parentheses along the ordinate, a x i s gre the experimental critical stresses in 10 kg/mm . ( c ) and (d) - D i s t r i b u t i o n o f s h e a r s t r e s s e s T., and T ( c ) and dislocation density (d) over the c r o s s s e c t i o n of a g a l l i u m a r s e n i d e s i n g l e crystal. 3  31  the  mean  RSS  shows  dislocations not  along  obtained.  2.  The  The  by  state  cylindrical a  the  wafer  effect  of  temperature  Newton's  were 13 9  ature in  Law  of  the  the a  the  •  7  1  4  •  8  1  GaAs  has  .  They  9  0  heat  flat at  the  cooling.  The  separately  as  determined  used  a  in  assumed  interface  surface  a  quasi-  equation  c o n d i t i o n s they  heat  is  analysed.  been  solid/liquid  the  symmetry  conduction  boundary  loss  stress  of  of  the  transfer  and  crystal  coefficients  function  of  temper-  fields,  the  stresses  191 .  the  Using  the  crystal and  crystal.  local  The  proportional absolute  the  values  of  are  plotted  on  four-fold  shown a  the  outside  The  principal  The  RSS  total RSS  was  acting  on  5.2  i s observed of  the  was {111}  a  more  radius  than  the  crystal  model  the  TRSS.  coefficients.  increases  oriented  sum  RSS  be  of  the  is  Typical between  directions.  follow  crystal  The  (TRSS)  midway  <110>  with  to  system.  crystal.  the  doubles  the  total  the  cylindrical  density  <100>  slip  along  the  strain  assumed  minima  of  dislocation  transfer  was  <110>  stress  c o n c l u s i o n s d e r i v e d from the  a  cylindrical  with  plain  d e f i n e d as  i n which of  a  for  density  This  plane  using  determined  RSS.  Figure  centre  Doubling cm  calculated  transverse  and  thermal  dislocation  in  symmetry  4  the  to  results  calculated  were  approximation,  2)  of  evaluated  '  1)  1  3  distribution  c o n d i t i o n s i s not  a1 .  at  heat  ;  i n growing  c o o r d i n a t e s . For  convection/radiation  used  et  shaped  diameter  field  Jordan  W-  growth  approximation  constant  following  typical  temperature  analytically steady  the  : from  increasing  2  to  heat  32  3)  The  dislocation  heat  transfer  density  coefficient  one  order  the  calculations.  The  temperature  3. thermal  of  stresses  relaxation  can  growth  is  small  is  is  determined.  during  reduced  s m a l l er  magnitude  field  be  the  calculated  included  the  if  the  estimated  as  values  used  numerically  in  in  zero  enough,  than  However,  to  this  and  case  analysis  in  the  stress  (Vakhrameev  192 et  al.  state  ).  The  thermal  including  thermal  field  conduction  crystal,  is  determined  equation  melt,  gas  for  and  by  the  s o l v i n g the  entire  crucible.  The  steady  growth  system  encapsulant  BO  19 3 is  neglected  solving  The  the  thermoelastic  displacement  stress  thermoelastic  field  is  equation  obtained by  a  by  finite  19 4 difference Figure  method  5.3(a)  Results  and  (b)  the  are  of  temperature  for  inclined  to  the  direction  of  the  dislocations  crystal  at  Figure  5.3(c)  stresses, axis  of  the  case and  and  are the  below  are  the  relaxed effect  of  Two  about  one  near 80  growth  and  well  formed,  other  vector  the  %  for  defined  1140°C. . The  on  considerable the  melting  parameters  a and  fraction of  the  temperature. thermoelastic  undoped  GaAs  are  along  observed  crystallization  are  discussed.  the where  front  stresses  conditions  i s not  In  as to  periphery  Thermoelastic growth  shown  perpendicular  regions  the  5.3(a-c).  is  densities,  CRSS  near  A  be  dislocation  Figure  density  <111>.  well  stresses  crystal.  Burger  to  shows  relaxed  a  observed  temperatures  dislocations 1238°C  is  in  dislocation  function  growth  shown  in  not  at this  given  33  Figure  5.2  T R S S boule.  contours  for  the top wafer  o f a <001>  GaAs  34  a  5.3  b  c  D i s t r i b u t i o n of the c a l c u l a t e d dislocation density in a gallium a r s e n i d e s i n g l e c r y s t a l b e i n g grown i n the <111> d i r e c t i o n for s l i p systems with Burgers dislocation vectors perpendicular (a) and inclined (b) t o t h e g r o w t h a x i s a t t h e f o l l o w i n g temperatures: 1) 1234, 2) 1200, 3) 1140, 4) 1050 C - T. (c) D i s t r i b u t i o n of c a l c u l a t e d dislocation density 1) t h e r m o e l a s t i c stress, 2) r e l a x e d thermal stress, 4) a n d c r i t i c a l s t r e s s f o r the formation of d i s l o c a t i o n i n u n d o p e d GaAs a t the corresponding temperature, 3) a l o n g t h e a x i s of a single c r y s t a l d u r i n g i t s growth ; CF) crystallization front.  35  4. both by  Dussaux  thermal  solving  subject . by  is  to  and  The  be  employed  using  is Von  element  thermal  method  field  the  to  was  equation  obtain  calculated  in  crystal  Mises  and  compare  generation  approaches  the  crystal  s u r f a c e as  Instead  of  Qualitative  can  done  energy  From  stress  are  not in  crystal  not  i n LEC  an  stress method  components in  crystal  are  for  (liquid extensive  The  the In  field and  the the  presented  lengths.  LEK  to  a  an Von slip  fixed  for model  two is  encapsulated analysis  of  growth.  have  the  plain  and  outside  resolved  the  different LEC  The  principle  the  boundary  coefficients.  considered.  contours  the  good  be  transfer  gradients  and  which  heat  realistic  gradients  thermal  stresses  in  more  temperature  Stresses  stress  and  dislocation  permits  minimum  derived.  Thermal  as  i s assumed.  layer,  to  chosen  dependent  the  methods  used.  The  C o o l i n g at  such  Kryopoulos)  assumed.  finite  conduction  geometries  diameters  5.  of  used  field  oxide  crystal  Law  scheme  different  stress  boron  heat  temperature  axisymmetric  system.  a  fields.  steady  numerical  calculated  Nises  employed  stress  Newton's  conditions  addition  has  . . 137,148,190 et a l .  The  crystal  and the  to  . , Jordan  195  been  plain strain  calculated  strain an  agreement  analytical  approximation  axisymmetric with  using  plain  is  approximation strain  and  not is FEM  19 6 calculations  is 19 7  approximation not  always  obtained  .  Using  the  same  the  plain  axisymmetric  198 '  valid  , in  i t was  shown  determining  that the  thermal  solutions  streesses  in  are the  36  crystal.  These  conclusions  are  obtained  for  semi-infinite  geometries .  199 6.  Galaktionov  approach that  to  the  residual)  and T r o p p  introduce  the e f f e c t  axial  gradient  are  numerically  produced  by  a  gradient  of the real  have  of  of axial  the  equal  hypothetical  considered  total to  stresses  the  method  They  In e q u a t i o n  form  assume  (transient  given this  by  and  stresses the  axial  i s expressed  as  9 0. . [ T ( x , y , z ) ]  3z  The  standard  thermoelastic  field  3  non  gradients.  temperature  field.  a  =  1 J  i s used  O  t  1 J  T(x,y,z)]  3z  to c a l c u l a t e the r e s i d u a l  stresses  in  infinite  cylinders.  5 .2  C a l c u l a t i o n of D i s l o c a t i o n  The  dislocation  function The  of the local  density  (MRSS)  or  relieved  may to  be  a  ways  Jordan  at a point  stresses  assumed  lesser  by p l a s t i c  different  1.  density  generated  t o be  value  flow.  Densities  i n a growing by  This  the thermal  proportional  i f the stress has been  crystal  is a  strains.  t o t h e maximum  i s considered  considered  to  RSS be  i n a number o f  :  et a l .  stresses  were  (model  2) a s s u m e d  partially  no  estimate  to  be p r o p o r t i o n a l  relieved  o f t h e amount  that  by p l a s t i c  relieved.  t o t h e TRSS.  the  The  thermoelastic flow  flow  was  b u t made assumed  37  2.  Vakhrameev  et  al.  thermoelastic  3.  Billig  in  stresses  (model  stress  1956  are  Is  and  3)  calculated  relaxed  by  Indenbom  completely  that  plastic  in  80  *  of  the  flow.  1957  proposed  that  relaxed.  12 9 Billig following and  estimated  simple  length  will  expand  bend  the  stresses  z.  analysis.  The  the  heated giving  are  relaxed  dislocations  are  and  vector,  in  Ge  of  this  the  a  by  n  Indenbom  The  =  temperature  gradient  involves  above a  =  a  =  thin  slab  of  temperature = az  6T. 1/R  n  is  using  =  the  thickness  gradient  This  deformation  6r  6T/6r  expansion  will  (1/z)x(6z/6r). a  given  number by  of  If edge  the  curvature  dislocation  densities  /cm  (6T/6r)  estimated 2  for  similar  (1/b)  given  expression This  [d(aT)  i s part  of  a  n  T  X  a  in  i s given  more  density  -grad  growth  conditions.  which  the  axial  by  /dt]  dislocation  =  density  therefore  (a/b)  4  a  radius  density  Billig  expression  tensorial  6r  is considered.  n  The  by  1/Rb,  3.2x10  proposed  a  plastic  relation of  of  curvature  formed.  order  Consider  surface  n  Using  dislocation  application  slab  Burger's  the  n,  complex given  by  equation  that  38  where  a  is  the  202  tensor  of  the  coefficients  of  linear  thermal  203  expansion  '  The  temperature  gradients  contribution to  the  of  total  the  radial  dislocation  and  density  axial is  not  estimate  the  S i , i s of  the  , 119,204 clear  Indenbom total  also  dislocation  proposed density  n  where  L  i s the  another  from  =  expression  plastic  to  deformation  e  e/bL  dislocation  l e n g t h which,  f o r Ge  and  10 6 order  of  0.25  introducing expression  mm. the  for  The CRSS  the  n  which  reduces  zero.  In  D  the  is  the  dislocation  to  the  are  calculated equation.  investigation  stresses.  a  glide  equation -y  in  values This  because densities  constant  is  from axial  2(T  given  Table the  equation i t  to  extented  obtain  '  the  by  following  _  and  for  plane  -  calculated  GaAs  were  density  i s the  diameter  on  equations  •+ (a/b)VTn  =  densities  N2  because  the  dislocation  crystal  measurements  dislocation  in  e q u a t i o n VTn  to  above  Indenbom  2 0V  cr  by  mean G  /Gb)(l/D)  Indenbom  axial is  with  the  is  thermal  columns  and  not  inconsistent stresses  shear equation  The  first  in  the  temperature  this  5.1.  when  second  used with the  gradient  in the  is  gradient,  modulus. are  The  compared  headed  Nl  and  terms  of  the  the  present  evaluation  crystal. does  CRSS  not  This  of is  produce  39  In extent  summary the  the reported  thermoelastic  dislocation  densities  pseudo-empirical laboratory  stresses  equations  which  In  densities the r o l e  results  adjustable  conditions.  dislocation gradients,  from  are  are not in  directly  of s t r e s s e s  relieved.  parameters  addition,  are  It i s not c l e a r  are  tested the  In  what  deriving  introduced  under  proportional  in  controlled  equations  i n generating  to  in  to  which  thermal  dislocations  i s not  clear.  Table  5.1  O b s e r v e d (N ) and C a l c u l a t e d exD Densities f o r Gallium Arsenide  VT |D,cm  N  |  | deg/cm  2  o  i  l i. N  C B  >  I N ,cm  |  9  Growth  method  |  cm  1,0 2,4 2,4 2,4 2,4 2,7  142 140 120 120 50 190  5 "3-5 ~l-2 "0,9 "5-6 5  0,8 0,8 0,8 0,8  450 380 290 210  "1 7-9 4-6 2-3  Note  n  ' 6X0  3  (N, and N ) Dislocation 1 * 2 Single Crystals.  10 10* 10* 10* 10 10 3  3  10 10* 10 10 5  1,7 2,0 1,7 1,7 5,7 2,7  10 10* 10* 10* 10^ 10  2,2 2,2 1,9 1,9 7,8 2,9  10* 10* 10* 10* 10^ 10  The C z o c h r a l s k i method  0,6 5,3 3,9 2,6  10 10* 10 10  0,7 5,9 4,5 3,2  10^ 10 10 10  Free zonal melting  4  5  4  and N were calculated with E q . ( 2 ) and Indgnbomjs equation, r e s p e c t i v e l y ; b = 4,10 cm | 2 1 ] , a =6,2 10 d e g and i t was a s s u m e d t h a t T /G - 1 10 . er g  40  CHAPTER  6  OBJECTIVES  In  the  dislocation are  previous densities  detrimental  technological industry of  to  require  the  the  GaAs  i t  has  levels  usually  preformance in  the  wafers  of  GaAs  of  been  low  pointed  present  LEC  GaAs  devices.  and  dislocation  that  in  fabricated micro  out  New  optoelectronic density  or  free  dislocations.  to  in  thermal  thermoelastic dislocation employ  distribution  the  basis  been  in  and  generated  as  grown  numerical and  distribution.  However  they  the  changes  in  in  limitations,  both  limited  models  growth  GaAs.  do  parameters  fields  on  This  plain  scope  growth  which  growing  of  not  the  models  crystal are  glide to  nor to  into  growth  based.  the models  models  symmetry  they  be or  can  of  the  analytical  numerical  take  These  reduce  the  mechanism  determine  can  mainly  describe  The  axisymmetric  Dusseaux  adequately  during  cannot  from  and  growth  crystals.  conditions results  the  a  four-fold  crystal,  strain  of  methods.  W-shape  in  during  developed  the  dislocations  The  have  derive  are  On  distribution  dislocation  the  GaAs  stresses.  analytical  develop  LEC  models  qualitatively  and  at  advances  Dislocations due  chapters  the axial  used  to  eliminate solution  approximations, model.  account  producing  Experimental  the  primary  the  thermal  measurements  41  of  the thermal  incorporate pressure  into  as a f u n c t i o n  t h e model  of the growth  are very  difficult  parameters,  to  make  in  to  high  growers.  The  main  develop  the  crystal  objective  taking  and  and  made  in a  to  have  the  a l l  determine  flexibility  axial  to  grower  i n the  o f t h e GaAs  will  be  The model  distribution  any  change  and  related  reported  i n the model.  incorporate  further  generation  quantities  dislocation  to  the variables  observations  pressure  conditions  the  of  and p h y s i c a l  high  is  for dislocation  account  fields  the boundary  able  investigation  Experimental  of thermal  establish  this  model  into  growth.  literature B O ,  of  mathematical  multiplication to  fields,  in  used  to  will  be  and  the  will  growth  conditions .  The  model  crystals fields  with  the  electrical  to assess  introduction  without behaviour  The sequence  used  growers.  f o r lowering  crystal  1.  be  i n the c r y s t a l  procedures the  will  or  of The  the d i s l o c a t i o n  modifications objective  eliminating  introducing  other  of  would  be  dislocation defects  density the  thermal  to  develop  densities  which  in  affect  in the  of the c r y s t a l s .  investigation  was  carried  out  in  the  following  thermal  stresses,  :  Based  on  a  mathematical GaAs  during  formulated,  glide  mechanism  models growth developed  for  to  dislocation  and and  due  generation  subsequent  validated.  cooling,  in  LEC were  42 Using  the  models,  dislocation growth,  on  role  of  the  dislocations  important  g e n e r a t i o n i n GaAs  were  Based  most  factors  crystals  affecting  d u r i n g and  after  determined.  above  thermal  results  and  the  stresses  in  i n LEC c r y s t a l s ,  were  thermal the  history  the  generation  of  clarified.  43  CHAPTER  FORMULATION  The  production  crystal the of  growth,  basic  stress  are  than  a  during  to  divided  ambient  from  the  thermal  for  both  parts  both  models  into  two  In  the  parts  process,  temperature.  separately.  of  from  the  are  then  Both  two  and parts  models  temperature  fields  the  which  the  glide  and  the  stress  critical glide.  thermal the  given stress  GaAs  {111}  <110>.  For  ten  when  twelve  different the  dislocations  particular  planes  in  a  specific  components.  component  of  or  crystal.  generated  and  directions  The  given  grown  in  of  planes  and  It  is  assumed  is  greater  RSS  are  the  planes  are  crystal  are  stress  Stress  directions  the  fields  The  the  ;  fields  stress  combinations  RSS  maximum  in  tensor  of  the  the  Mises  point  7.1  identical  fields.  Von  and  value,  The  growth,  planes  are  that  thermal  the  the  Figure  way  those  from  in  are  during  the  derive  process  i s the  a  shown  case  direction,  model  of  at  is  our  system  there  the  calculate  (RSS)  projections In  of  time  to  models  fields  calculated  used  Stress  directions.  the  made  specific  and  in  cooling  a  are  directions  solidification  chart  Shear  [001]  the  flow  At  RSS  the  includes  the  between  derived  the  be  2  calculated.  Resolved  can  that  chart  components  MODELS  decoupled ^®.  fields  difference  THE  crystals  modelled  is  simplified  flow  are  are  assumption  A  The  crystal  process  fields  by  which  subsequent the  of  OF  7  and  multiply  i n which  the  44  TEMPERATURE FIELD  STRESS  V O N MISES  FIELD  R E S O L V E ;D S H E A R S T R E ;ss  STRESS  MAXIMUM RESOLVED S H E A R S'rRESS GLIDE  MODE  N O DISLOCATIONS  Figure  7.1  Flow c h a r t fields.  of  the  model  used  to  derive  the  stress  45  RSS of  are the  one  in  other  A  of  crystal  during  is  The  The heat  of  to  relaxed  also and  crystallography  assumed  there  mold. and  temperature  and  solution  i.e.  =  shape  remains  as  well  The  heat  at  field of  the  as  is  that no  when  further  the  the  LEC-GaAs problem of  phases,  the  mechanisms  made  this  crystal,  and  treat  are  a  solution  mold  To  assumptions  in  Stefan's  four  transfer  in  a  exact  includes  field T  The  convection.  following  interface  the  and  melt  radiation the  temperature  conditions.  and the  the  requires  dependent  symmetric,  2.  is  and  Equations  boundary  mathematically  1.  is  It  mode  growth  growth  encapsulant  conduction,  generated.  stress  analysis  time  input/output  glide  Field  Governing  complex  problem  being  the  directions.  complete  crystal  determine  the  Temperature  7.1.1.1  gas,  operates,  Model  7.1.1  with  will  dislocations  mode  glide  7.1  maxima  heat include  this  problem  is  axially  :  crystal  T(r,z,t)  is  unaltered  the  melting  by  dissipation  point,  of  latent  T M P  3.  Heat  transfer  the  solidifying  Cool ing  at  the  interface  interface)  of  the  follows  crystal Newton's  (excluding Law  of  46  4.  The  heat  transfer  allowed i.e.  to  in  vary  coefficient  along  contact  the  with  and  ambient  temperature  lateral  surface  of  the  encapsulant  the  or  are  crystal,  the  gaseous  in  crystal  atmosphere  5.  The  temperature  length  with  field time,  is  changed  i.e. a  by  a  change  quasi-steady  state  approxim-  ation.  With heat  the  above  conduction,  solidifying  assumptions  with  respect  interface,  as  33 2  (6)  L  =  3p  the  to  shown  a  -r- +  coordinate  3?  equation  of  fixed  to  the  the  form  :  , takes  36  2  +  system  7.2  36  3p  p  dimensional  in Figure  36  l  non-  r - - 2V  3£  =  o  (a)  (7.1)  p = T  where  2V  crystal  =  v  r  -  MP  Q  /  T  v  ,  Q  £  =  z/r  (b)  Q  0  K , T  radius,  r / r  Q  is  the  reference  the  growth  velocity  also  satisfy  the  temperature, and  K  r  Q  is  the  the  thermal  general  boundary  conductivity.  Solutions conditions  i.  must  :  Dirichlet At  following  the  condition solid-liquid  revolution  6 =  f(r,z)  1  =  0  interface , r  <  given  by  the  suface  of  r^  (7.2)  lz=0  r  melt  Figure  7.2  C r y s t a l c o n f i g u r a t i o n and the m a t h e m a t i c a l model.  c o o r d i n a t e system  used i n  48  i i . Neumann At  any p o i n t  33 (— 9  condition  33 n + — n )_ P £, S D  p  3  z  crystal  at  coefficient  Finite  equations  solution small the  the  between  In  scheme  ;  of the  at a distance  h(z)  crystal  (7.3)  n  normalized  temperature  is  surface  z  the and  heat  ambient  from  the  of the transfer  ambient  at the  finite  this  S. . M 3  approaches  element  method,  after  the element  e  )  )  (e)  N <  this  i s given  e )  i  D  the  each  the  the Galerkin's whole  3 i s required  equation  over  d  f o r obtaining  method,  variable  integration  (  to solve  below.  Equations  f o r the f i e l d  following  employed  different  elements,  D  point  crystal,  - 3 U ) l a  temperature  surface  i s the  g  that  Element  in a  selected.  3  of the  point.  numerical  Among  i s the normalized  and  surface  -h(z) r.rfi 0 S  to the c r y s t a l  interface  7.1.1.2  =  ?  ( )  adjacent  same  n  F  3  where  The  S on t h e c y l i n d r i c a l  domain  element  method in  which  is discretized  i s obtained  by  was a in  performing  element,  0  ( e )  with  (7.4)  (e)  [N]  {3}  (e)  [H  2  N  0  . . N ] < g ^  g  49  where  the  nodal  field  of  N's  are  the  values  interpolation  i n element  functions,  (e) and  g  3 ^  the  i s the  degree  are  of  the  freedom  the element .  The  interpolation  continuity  of  up  order  to  one  operator. problem parts  and in  functions  at  first  the  use  such  a  II  Equation  variable  than  means  can  Appendix  field  less  This  are  problem  the  functions  of  to  that  i t s partial order  required.  However  show  less  These  strict  calculations that  transformed  the  into  are  initial the  is  derivatives  functions  conditions  that  there  derivative  boundary  i t i s shown  is  of  such  interpolation  the as  and  be  highest  apparently  used.  where  (7.4)  that  way  be  the  must  i n the  for  a  C*  integration transform  by the  interpolation presented  problem  following  in  given  by  equivalent  equation  3N  3g  3N.  ( e )  ( D  + 3P  (e)  +  =  r h(z) Q  B  N. (  e  (  6  3g  )  3  (  E  )  ( e )  + 2V N.  3£  f s  3  -  9P  r h(z)  3  9?  dS  (  e  )  ) dV  +  (e)  35 .  1  =  )  (z) f N.dS J (e)  (7.5)  ( 6 )  s  where  S^ '  elemental  e  i s the surface.  interpolation  surface It  functions  area  i s now  of  clear  f o r C^ ^ 0  element from  problems  (e)  the are  and  above  dS  i s the  equation  required.  that  50  A  suitable  axisymmetric  element  bodies  shape  is  chosen  the  here  for  axisymmetric  subdivision  ring  element  of with  triangular cross-section.  The  field  element  and  triangular problems are  a  (  are  P,  three  toroidal  defined  L.  variable  the  is  assumed  to  vary  node  element  is  chosen.  elements,  n a t u r a l or  linearly  interpolation  area  coodinates  For  inside these  functions function  the  linear  for  C^°^  . The  L. 's  as  (a  C)  2 A  + b  i  p + c  i  )  (7.6)  where  a i.  where are  PjSc  =  " k j P  ( i , j , k)  the  ?  have  coordinates  numbered  i " k  -  module  3  :  c  of  P  the  p  j  a  and  nodal  n  d  b  i =  permute  points  ?  j " k ?  ( 7  cyclically,  (vertices)  of  -  7 )  ( Pj , C a  triangle  counter-clockwise  2 A  A i s the  The  (7.8)  area  of  the  temperature  triangle  within  this  element  is  expressed  by  the  equation (e) 3  (e)  ^  (e) [ L  1  L  2  L  3  ]  <! 02  (7.9)  51  where (e)  3 ^ ' s are the nodal  the  i n the  local  from  and  3^  UK ]  +  X  ^  6  considering  1 t o 3 we  may  [K  ]  3x3  of the temperature  In  element  numeration.  Substituting (7.8)  values  given  that  write  < 3 >  )  3x3  (  by  the  Equation  integral  Equation  E  )  =  (7.9)  should  be  into  performed  (7.8) i n m a t r i x  {K  Equation  form  for i  as  (a)  )  1x3  1x3  where  { (e)  ij  3  3L  9L  3L.  +  1  P  3  (  3C  P  3L. ) dD 3C  + 2VL.)  1  1  1  . 1 6  '  (b) (7.10)  L.L . dS ( e ) H  K  i j  °  A. . ij  The  =  (e)  T  is  their  axis  of  = i j  A  [K.] A  [K ]  l  surface  i) vertical,  J  A  a  r  e  elements with  1  >  1  6  , . [3  p  ( e )  + p.] J  according  respect  elements.  + c.c.) +  calculated  i i ) tilted  TTVb. 1  i n Appendix I I I .  3  K  peripherial  the external ;  i s given  and  H  different  e )  (b.b. 2  exactly,  on  (d)  1  n  crystal  peripherial  (e)  , [K„] and  I  f o r three  the  Trp* T  (e)  [K„]  orientation  horizontal  K  L . dS  calculated  approximately to  of  (c)  J  r h ( z ) B (z) o a  evaluation  [K ]  i  to the  and i i i )  52  where  i, j  p,  (1)  [K ]  =  H  =  *  1,2,3  P  * P  2  Vertical  2Trh(z)r  3  boundary  ( )  Pl  l j  1/3  1/6  0  1/6  1/3  0  0  0  0  (7.11)  1 <K  A  }  =  Trh(z)r  0  B  a  (z)pl  1 >•  1j  0  (i1)  Tilted  27Th(z)r  Boundary  3P.  +  p.  P.  +  P.  +  P.  P.  +  hi  (  P.  0  3 p.  0  12 0  1 TTh(z)r  0  B ( z ) (p a  i  +  2 p )  Al  1 0  53  (Hi)  Horizontal  boundary —  The length  same  as  of the element  Details coding,  of  type  7.1.2  To  i.  7.1.2.1  To  was  (u,w)  =  matrix  1^  i s the  including assembly  computer  and  matrix  from  of the  The  and  crystal  the  isotropic  traction ,  n  at the surface, i . e .  i s a vector  perpendicular  crystal.  Equations  equation,  strain  the  i s axisymmetric  o x n- = 0  the element  in  :  elastic  and Element  used.  stresses  field  i s free  the surface  obtain  defined  t h e minimum  energy  for a  potential  linear  elastic  energy solid  as  [e]  1 2  where  is linearly  Governing  principle  p  thermal  a r e made  the surface  to  '  v  later.  the  the c r y s t a l at  calculations  refinement,  the thermoelastic  i i i .  p  boundary.  numerical mesh  pby  replacing  e  Model  assumptions  t h e body  ii.  for  Field  calculate  following  U  the  are given  Stress  but  at the  o f mesh,  calculations  is  in (i)  ( )  {a} dv  V  (a) i s t h e s t r e s s  the present  (7.12)  case  tensor  have  and  [e] i s t h e s t r a i n  the f o l l o w i n g  components  tensor  which  54  (7.13)  { o>  TZ  The  stress,  constitutive  equation  that  [EQ] i s t h e s t r a i n  solid  due  to T  coefficient  of thermal  . This  Q  strain  strain  with  f o r an  expansion  Duhame1-Newman strain,  is  expansion  respect  isotropic  to  of the  reference  material  with  a  a is  (7.15)  1 0  —  (1-2V)  E t h e Young  to a free  'differences  c o m p l i a n c e mat r i x w h i c h  (1+V)  with  f o r the thermal  corresponding  E  [C]  the  a  V  a( T  the  and  (7.14)  temperature  temperature  is  accounts  Law  0  where  [C]  t o Hook's  ({e} - { e > )  [C]  io)  according  modulus  c a n be d e r i v e d  from  1- V  V  V  1-V  V  V  0  0  and  v  for  this  V V  1- V 0  case  0 0 0 1 -2v 2  the Poisson's  the displacement  ratio.  {d} a s  (7 . 1 6 )  T h e { e)  55  ie  (7.17)  )  where  [B]  i s the  following  matrix  of  derivatives  3r J r  [B]  (7.18)  0  "3~i~ 3 3r  _3 3z~  Taking  into  displacement,  II(u,w)  =  U  account  the  the  potential  (u,w)  relation  energy  [d]  [B]  among  can  be  [C]  [B]  stress,  written  strain  and  as  {d} (7.19)  2[d]  When assumed field  the  that  body within  i s approximated  is each  [B]  [C]  subdivided element  ,e  Q  into with  g  dV  small  elements,  nodes  the  i t  is  displacement  by  <;> = te!} ' = { i ^ O = e ,)  {e }  (e)  [N],d,  (7 . 20)  56  where  [N]  are  the  interpolation  functions  matrix  of  order  g  and  (e ) {d}  are  the  element  minimization  process  result  force-displacement  the  obtained 2gx2g [ k ]  (  e  {d}  {d}  ( 6 )  =  {F}  out  element  equation  for  this  by the  case  element. element  the As  a is  ( 6 )  (e)  {d},  w.  (e) (7.21)  w. {d}  2gx2g (e) [k]  carried  In  2gxl  {d> (e)  be  displacements.  as 2gxl  )  may  nodal  (e) g  w  g  [k]  (e) 11  ™[V  ^  (e) In  [k]  (e) il  [ k l j ^  tk]  (e) ig  [k]  (e) gl  t k ] ^ ^  [k]  (e) gg (7.22)  2gxl (e) (F)  {F>  (e) 1  {F}  (e) 2  (e) {F} g  57  In  the  stiffness  stiffness  matrices  at  / [B]{  2  x  matrix node  4  m  (e) [k] i j  (  e  )  [K],  i and  the  submatrices  have  the  x 4x4. . 4x2. . [C]< > [B]< > e  [k]^j  are  the  form  dV  e  (e)  3N 3r  N  [B]  (7.23)  (e)  3N 3z 3N  3N ,  3z  The  [1 F  }  initial  force  2x4 [B]\  ( C )  O' i  (e)  (e) {FQ^  matrix  . .4x4 [C]  4x1 . . fe ]{ e  0  ,  l s  (7.24)  dV  (e)  7.1.2.2  Requirements  Since the  3r  only  integrand  interpolation  first  for  the  function  for  the  order  interpolation  derivatives  potential with  energy, continuity  of  functions.  displacement  compatibility of  appear  in  requires  an  displacement  at  the  58  element  boundary  Interpolation  functions  displacements is  and  uniform  that  elements  used are  matrices,  dimensional  2x2  quantities  . . 2x1  fk'lij  for  node  states  represent  within  and  C^ ^  first  linear  temperature  or  of  higher  obtaining  equation  an  is  . (e) }  2x1  field. are  in  (  E =  {  a  )  -  MP  T  d  }  (e)  ( f e )  ) r 0' l0  {k')!^  =  [B']<  =  r  e  [B']J  6 )  PdPdC  (c)  where  [B][  (1 + V)  [C]  (a)  e)  Q  that of  such  as  This  the  means  suitable.  terms  form of  of non-  . .  ( ) / [B']j [C]  e )  body  satisfied  specific  written  <V>i  =  a{ T  is  terms  where  { d  rigid  derivatives  order  the  the  element;  continuity  1  the  completeness,  as  .  {d'/j  to  containing  before  the  able  This  linear  convenience,  element  for  displacement  functions  which  be  strain  respectively.  coordinates  For  must  of  interpolation  natural  However,  constant  states  displacements with  only.  ( 1 - 2 V )  =  [C] E  (b)  (7.25)  59  { F i r  -  e  7.1.2.3  6  Formulas  The using  T  >< > O'l  stress  the  / [B  d  for  is  V  [C')<  Stress  tensor  following  (7.26)  (e)  ]J i  1  pd pd C  (c)  Calculations  calculated  at  each  node  i  in  element  e  equation  ,(e)  1-  V  V  o.  V  V  1-V  V  6.. 1  > - <  > v  o.  v  1-v  nodal  components  displacements  ie'}  For  strain  -  {e)  nodes  at  the  at  1-2  each  V  node  are  calculated  obtain  stress the  [B ]! ) 1  <d')!  6  axis  of  the  employed  calculated  stress  from  the  as  (7.28)  e )  body e  Q  is  substituted  for  0 The  (7.27)  o  0  The  V  in  in  absolute  this units  e  .  p way the  is  non-dimensional.  following  relation  To is  60  {0>  E a (T  (e)  T  MP  ) (7 . 29)  (1-2V)  7.1.2.4  The vertices elements  Linear  Elements  linear  elements are t r i a n g u l a r  of the t r i a n g l e . are  displacement  {d}  the  natural  i s written  (e)  L  L  Making terms  [  k  ,  ,  i j  The  +  1 1  +  W  coordinates  L  L  2 2  +  2 2  +  U  W  L  L  the corresponding  of the s t i f f n e s s  interpolation ,  nodes  functions L  0  and  L  and t h e  U  (7.30)  3 3. W  calculations  and i n i t i a l  11  12  21  22  force  and s u b s t i t u t i o n s ,  matrices are  (a)  }  [(1-v)  b.b. + I j  P  +  v  ] — 4A  c.c 2  J  ( I - V ) A ;-  + 6  [b.+b.] + J  dpd?  (b)  P  ( 1 -2 v) 12  [ vb . c . + i J  atthe  f o r these  3 3  ( 1 - 2 V) 11  L  with  as  1 1 U  toroids  P c  iV  v +  4A  6  c . J  (c)  the  61  (l-2v) [ vb  21  c J  p  +  c 2  1  J  b  V  ]  + 4A  1  c  (d)  6  (l-2v) [(l-V)  22  C^Cj  +  b  l  b  j  ]  (e)  4A (7.31  (bj  2 A  p +  (f)  F'  (g)  An terms  Inspection can  which  be  should  L  =  be  for  calculated  the  derived  back  stresses The  at  stress  nodal  the  Quadratic  of with  a l l elements  quadratic three  except  the  average  only. and  Once  term  the  each  node  sharing  the  a l l  containing  of  p  and  displacement at  are  each  is  the the  the 1/p  intermediate  corresponding  element  that  One  values  the  stresses  in the  shows  numerically.  taking  term  centroid  7.1.2.5  and  of  strain  associated  relations  exactly  substitution  from  vertices  above  integrated  that  stresses  The  the  consists  1/3)  by  of  evaluated  approximation and  )  (~p  field  element  can  equations.  is be The  average  stresses.  average  value  of  node.  Element  element nodes  at  has  six  mldsides.  nodes,  three  Interpolation  nodes  at  the  functions  are  62  built  from  stiffness  to  includes  node  that  elements  the  derivatives instead  interpolated  coordinates.  matrix  note  only  that  the  natural  and f o r c e  important  at  the  of  [B]  the non-zero  i n the  the displacement  of  terms  of  element.  components  calculation  i n Appendix  matrix  the d e r i v a t i v e  displacement  element  detailed  i s given  nodal  of  A  these of  a l l the More  V.  of  It i s  elements  displacement terms  i n the  specifically,  are interpolated  as  in  usual  as  u  i  \  i  e  Uj  where  Z j=i  and  interpolation  l  e  N.u. 3  w..  3u  (  a  true  =  quadratic  the  (7.32)  £ N.w, j=l  nodal  J  displacements  F o r i n s t a n c e , by  6 Z j=l  )  3r  and  6  =  J  are  e  \  6  3  =  ;  w* '  functions.  , . e  i  6  =  ;  and  N  the  definition  3N. (7.33)  J— u  9r  element  3  will  have  a  B  element  at  node  i  which i s  3N.  6 B  1  (  1  i  £  —  j=l  3r  (7.34)  N.) J  N,  = 0  NJ.-  However term  i n the pseudo-quadratic  N^.  1  J  *  elements  1  employed  here,  only  the  u^ i s c o n s i d e r e d , i . e .  3N B  i  =  U  (7.35)  l  or  63  It  is  stresses  also the  full  representative  7.1.3  Von  and  Mises  The the  important  these  quantities  that  stress can  be  material.  This  Von  Mises  Resolved  be  compared  Stress  for  The  G  V M  =  «J  The stresses  The on  the  is  formed  to  GaAs  Von  to  Mises  1/2  [<  0 l  principal that  are  1  slip  planes  by  the  quantitative a  should  with can  the  be  Stresses stress  described (RSS)  and  and  '  +  (a  stresses,  by  the  obtained and  {111}  to  twelve  slip  RSS  a  3  )  the  +  2  (o  directions.  as  <110>  well  components  generation  terms  these  into  of  values  of the  must  Resolved  Shear  generated.  principal  2  the  stresses  by  -  o ) ]  (7.36)  are  the  values  of  local  stress  tensor  glide  system  2  3  the  tensor.  resolving  and  in  are  definition,  stress  by  planes  procedure  ~  1  to  of  deformation  Critical  related  with  transformed  plastic  is  2  of  obtain  description  be  Stress  2  to  obtained  dislocation  if dislocations  0 )  order  coordinates  For  field  calculation  stresses.  determine  The  the  Stresses  associated  yield  combinations. obtain  Shear  components  diagonalize  RSS s  of  a  the  in  values  gives  Shear  for  used  cylindrical  stress  the  that  is  stresses.  the  and  in  method  induced  analysis,  matrix  Resolved  tensor  element  thermally  note  consistent  and  stress  finite  [B]  to  as is  the In  GaAs  the  directions the given  giving  detailed in  12  slip  calculation  Appendix  VI.  64  The are  twelve  given  stress  in  Table  level,  combinations shear  RSS  a  7.2  at  of  stress,  for  crystal It  every  planes  these  can  grown  in  seen  that  be  point  and  in  the  that  direction  independent  space  directions  [001]  there  have  the  of  are same  the  three  resolved  are  (111)  [110]  ( T i l ) [HO] ( T i l ) [110] This  leaves  "glide  modes".  the  sign  are  listed  of  7.1  the  for  crystal  are  used.  of  Substituting results there  shown are  corresponding  The the  RSS  stress  analysis  the  in  normal  stress  will  labelled  stress.  be a  called  and  The  and  ten  stresses  the  that  For  b  is  modes  the  It  be  seen  can  different  of  the  or  is  higher  modes  that  stress  axis in  Table  chosen  in  dislocation  planes ten  produce,  sections  given  plane  these  for  to  as  (010)  to  two  parallel  equations  7.2  five  difficult  distribution  evidence plane.  given  are  sections  sections  that  Table  only  shear  s e c t i o n s normal  experimental  in  modes  contours  of  parallel  in  the  which  7.1  For  occur  levels  between  involving  used,  For  the  densities  in  the  stress  difference  term  direction.  are  view  The  different  dimensional  therefore  growth  ten  in Table  Three  the  only  in  levels  8  =  0.  gives  the  this with  plane the  multiplicity.  are  compared  literature  for  with  different  yield,  critical  dislocation  stresses generation  given and  65  multiplication  f o r doped  and  undoped  GaAs  as  described  in  section  8.1.  Table  7.1  Resolved  Plane|Direction|  [110]  Shear  Resolved  /6  [Oil]  Shear  Components  (001)  Crystals Node  JCOS (  q  P  in  Stress  {-(a - O ) C O S 2 6 + /2T  6  (HI)  Stress  y  8 + TT/4 ) }  0  /6  { o - a J / 2 c o s 9 s i n ( 9+TT/4 ) - ( o - o ) s 6' P e  /6 ~~g  /6  g  sin6}  Q  r  {(o  P  p C  Q  0  IV  a  V  b  - o j  _cos6} PC  /6  [Oil]  j/6 { - ( a - O o ) / 2 s i n 9 s i n ( e + Tr/4) + ( o - a  (- ( a - o ) c o s 2 6 + / 2 T ^ c o s (6 + 7T/4 ) } p  Q  e  P  -T  ?  c COS0}  f i  p ?  g  {-(o  P  -o  r t  )cos26+/2T  6  p?  )  -  e  /6 ( ( a -o„)/2cosesin(9+77/4)-(a^-o.) -gp 8 C 6 +T sin9}  /6  I a  p  r  6  [110]  a  C O  [110]  [101]  III  a  sin9}  - o ) / 2 s i n 6 c o s ( 9+7T/4 ) + ( o  -T  (111)  p ?  { ( a -0 )/2cos6cos(e +7r/4)-(a -o ) P 0 t, t) -T  [Oil]  a  -  Q  -T  [HI]  II  0  p  [101]  a  / 6 {-(0 - o ) / 2 s i n e s i n ( 6 + Tr/4) + ( a _ - o ) + ~g" P o t, a + a £.cos6)  [101]  I  per  II  b  III  a  I  a  +  COS(9+TT/4)}  Cont./  66  Cont./  [Oil]  /6 { ( a - o ) / 2 s i n 6 c o 8 ( 8+TT/4) + ( o _ - o ) + p 0 t, 0 6 + T c o s 6} p£ Q  Q  r  (111)  [ToT]  j/6 { ( a - o J / 2 c o s 6 c o s ( 6+ n/4 ) - ( a - o.) + 6 P e c e + T .sine) p£  [110]  {-( 0 - 0 ) C 0 S 2 0-/2 T ,,008(9+^/4)} p o pt, Q  6  V a  IV  b  I b  67  7 . 2  Table  |  Node  1  Resolved Shear S t r e s s a [001] C r y s t a l  RSS  A  2  |  Components  P r e v i o u s Modes  5 6  7.2  [ o -oJ P K  to  the  separate  temperature. not  crystal  been  The  i t  has  from  effect  established  (111) [ 1 1 0 ]  11  a  V  a  (111) [oil] (111) [ 0 1 1 ]  11  b  (111)  [Oil]  V  b  (111)  [Oil]  111  a  (111) [ 1 0 1 ]  III  b  (TTi)  [ToT]  IV  a  (Tii)  [ToT]  IV  b  (ill)  reached  the  |  [10T]  growth  melt  of c o o l i n g in  | Direction  b  After  the  Plane  I  ]  Modelling for Cooling  Once up  O - O J - T  |  a  6  [(  (010) Plane f o r  I  3  jM 6  a  (111) [ 1 1 0 ] (111) [HO] (111) [ 1 1 0 ]  6  /6 6  4  in  i t s final and  length,  initiate  i t is  cooling  pulled  to  in generating dislocations  literature.  However,  the  room has  cooling  68  practice  may  induce  dislocations,  In  particularly  order  dislocations developed.  based  on  to  study  the  solution  the  was  evaluations  justify  of  the  i n the  the  are  use  of  end  of  The  has  not  expensive  and  generating has  temperature of  this  easy  to  been  in  approach  are  to  been  field  an  method  was  obtain  numerical  and  methods,  sufficiently time  more  region.  selection  compared  generate  cooling  dependent  Solutions  cheap  cooling  tail  will  mathematical  time  a)  hot  that  effects  chosen.  factors.  importance  stresses  following  obtain  two  numerical The  to  To  analytical  thermal  b)  established  consuming  numerical  methods.  The  analytical  following  temperature  assumptions  1.  The  2.  The  were  calculated  making  the  :  temperature  crystal  fields  field  is a  i s time  finite  dependent  cylinder  and  axisymmetric.  with  constant  (gas  or  cross-  section.  3.  The  crystal  constant  4.  Law  the  transfer  heat  of  determined  surrounding  The  initial  dependent  in a  medium  liquid)  at  a  surface  ;  temperature.  Newton's  is  5.  i s immersed  Cooling  applies  coefficient  by  the  at  is  the  crystal  constant  temperature  and  and  i t s value  nature  of  the  medium.  temperature  (radial  of  temperature  the  crystal  gradients  is  during  axially growth  are  69  small) fits  the temperature  the With  ; and i t i s g i v e n by  moment i t i s p u l l e d  these assumptions,  following partial dimensional  quantities  6  the melt. fields  s a t i s f i e s the u s i n g non-  33  2  (7.37)  ) + 3p  3t  (K/r *)  t  0  Condition  t = f(P)  0  P  +  0  Pj  S  t  >  P  +  Boundary C o n d i t i o n s , w i t h h at  that  of the c r y s t a l at  e q u a t i o n and c o n d i t i o n s  3 3  (P 3p  Initial  from  the axis  function  as b e f o r e :  33  where t  at  parabolic  the temperature  differential  3  at  field  a  1  2  C  = hr  (7.38)  0  0  33  3p  -h'  3  (a)  •h'  3  (b)  P =1  33  3C  33  3C  5  h =o  1  3  (c)  (7.39)  70  The  solution  to t h i s  problem  a s shown  i n Appendix  V  i s given  by 2 * 3(p.£.t*)  e~  2 h ' [ Z CAX)  =  X  X  J.(Ap)]  1  1  0  2 * {£ Y  where  the  2  \-  following  <Y)  Y  C  t  e  values  algebraic  AJjU)  t a n  C  Y  [YcosyC  1  y -  and  + h'sln Y q  values  are  (7.40)  the  solutions  equations  =  h' J ( A )  (a)  =  —  2y h' -j- h  (b)  0  y  (7.41)  1  with  Cj  <M  =  ~  (h'  (a)  ~  2  2  A  d +  )J (A) 0  and  11 C  II  =  2  ( )  -  Y  siny^  (b)  {p  +  Q  P  (C  j  2 + 2 C  t  + P (2? 2  h'/Y  t  "  h  C  }  '  of the  O  S  + h'/Y )  y  2 t  <-P  +  2 n  2  2  -  2/Y ) 2  h  +  C  + P (C  2  t  '/Y } 2  2  P  + -  £  h 0  0  '  h  — Y  +  '  l  p  P  - -  (  1 1  Y  1  " ' h  C  t  ? ' - — f Y 2  p  )  n  (c)  71  (7.42) €t  The to  ,2  (Y  temperature  calculate  the  ,2, h' ) +  L  +  finite  stresses  the  field  temperature  (d)  calculated  thermally  element the  h  method  Von  Mises  dependent  induced  RSS  critical  a  given  stress  described and  at  in  are  values  time  field,  with  section  7.1  derived  and  for  is  yield  employed  the  .  aid  From  these  compared  or  of  to  dislocation  g e n e r a t i on.  7.3  were  Analytical  Solutions  Analytical  solutions  obtained  compared  with  analytical boundary  7.3.1  and  numerical  those  the  temperature  evaluations  obtained  solutions  with  developed  of  finite  were  and  stress  these  solutions  element  obtained  fields  methods.  for  were The  simplified  conditions.  Analytical  The solved  for  quasi  QSS  -  Temperature  steady  analytically  following  for  c o n d i t i o n s are  1.  Planar  solid  2.  Cylindrical  state  -  a  differential  simplified  adopted  liquid  crystal  Field  equation  problem  in  :  interface  with  constant  cross-section  (7.1) which  is the  72  3.  Uniform  heat  transfer  temperature  4.  Constant  From  T  the  h  and  constant  of  the  crystal  ambient  fl  temperature  these,  coefficient  at  boundary  the  ends  c o n d i t i o n s are  written  as  "  93  h  1  3  (a)  s  p=l  9p  31 <5=0  =  31  = 0  1  (b)  E  ;  =  ^-  (7.43)  (c)  0  where  i s the  The to  the  solution  Q  and  to  specified  equation J  total  in  length  the  heat  boundary  term  of  of  the  the  crystal.  conduction  conditions,  equation  is  Bessel  function  ( ^P)  te  given of  zero  -k C  =  2hr  J 0  e ^ Z  following  first  order  values  A Jj(A)  =  are  the  hr J (A) 0  Q  k C  T  e  x  e ]  r/)J (A) n  0  the  x  -  ;  v  A ( A +h  and  the  and  -2k C  x  where  by  subject  Jj  v£ g(p. O  (7.1),  (1  0  solutions  of  the  -  -2k, e A  (7.44)  c_  M  algebraic  equation  (7.45)  73  k  v  .  2  x  2 +  A  Details  of  7.3.2  the  Analytical two  (7.46)  calcuations  Analytical  using  2  solutions  solutions  different  axisymmetric.  are  Two fields  temperature  gradient)  for  for  the  i n Appendix  Stress  the  of  that  depend  and  field  :  temperature on  were  plane  fields radius  general  VII  Field  stress  approximations  types  temperature  given  obtained  strain were  only  employed, (no  axisymmetric  and  axial  temperature  fields.  7.3.2.1  In field  Plane  this  Strain  Approximation  approximation  consists  of  primarily  component  constrained  ends  a  of  cylinder.  temperature components  oP  2hr  oe  2hr  o  2hr  field of  to  i t a  is  radial  maintain For  hr  that  the  component, axial  two  dimensional  in stress  section from  Jj(Xp)  displacement  with  zero  the  calculated  non-dimensional  assumed  traction  at  the  IX  the are  1 J (A)  Xp  ]  (a)  0  (  Jj(X  P)  axial two  axisymmetric  7.3.1,  Appendix  the  - J (XP))] ( b ) 0  Xp  (c)  three  74  where  (7.47)  e K  x  (C)  a  =  - e  -P  A  -  2 k  *t  A  k  e  A  5  (p  OQ  =  1/4  (3p  0  =  p  7.3.2.2  -  2  of  stresses  necessary  by  (a)  (b)  a  i n Appendix V I I .  Solutions  using  Love's  and  particular potential  stresses.  satisfy  (7.49)  (c)  Love's  are  given  1)  the  i s a c y l i n d e r with  Solutions  )  the f o l l o w i n g r e l a t i o n s h i p s  solutions  elastic to  2 Q  1/2  gives  isothermal  by  -  2  using  while  r  the c a l c u l a t i o n s are given  Analytical  potential  field  2  - 1)  2  Axisymmetric  obtained  h  (7.48)  1/4  ?  +  2  2  =  p  ) ( \  l  are given  Details  body  - e  temperature  stresses  0  were  ^  (d)  radial  3  the  k  = (1  For  "  obtained  A  the  Goodier's solution gives  using  a  the  general of  In  c r o s s - s e c t i o n and Fourier  Goodier's  thermoelastic  solution  both  conditions.  finite  approximation  potentials. for  combination  boundary  constant  axisymmetric  f o r the  potentials is this finite  case  the  length.  transforms  for  75  axisymmetric  and  radial  temperature  fields  as  i n the p l a i n  strain  case .  Details Appendix  For  of  the  calculations  and  solutions  are  given  in  IX.  purposes  non-dimensional non-dimensional  of  comparison  stress  component  stresses  approximation  and  approximation  with  v  by,  b)  (1  =  0.29  -  with  the  were  obtained  a) 2v)  1 /  (1  -  finite  2v - v )  element  by  for  method,  multiplying  the  f o r the  the  axisymmetric plane  strain  76  CHAPTER  EVALUATION  In the  this  analytical  addition method  the  8.1  compared  of  Programming  Based programs  on  in  charts  and  are  the  stresses  the  same.  program  a  different  meshes  to  the  A of  Choleski  equations  the  case  for  the  of  the  last  obtained  with  was  the  compared  The  finite  model  with  chapter.  measurements  which  method  the  and  written  GaAs 8.1  In  element  in  a  high  is evaluated  on  was  temperature the  for  the the  last  calculation  during  8.2  chapter,  growth.  The  flow  of  Flow  charts  programs  employed  for  calculating  quadratic  elements  are  essentially  as  the  field  in  elements  best  selected from  generated  shown  triangular gave  in  crystal  and  automatically  resulted  of  presented  were  in  chart  using  selection  scheme  computer  linear  that the  in  growth.  Figures  flow  one  are  Parameters  language  the  mesh  with  select  Input  in  with  The  during  stresses  to  models  temperature  numerical  shown  corresponding  the  results.  FORTRAN  temperatures  MODELS  presented  with  and  the  of  fields  puller  these  THE  results  solutions  Melbourn  basis  the  temperature  are  pressure the  chapter  OF  8  Figure were  solving finite  calculations,  solution  a  computer  8.3.  Several  tested  in  order  results.  for the  using  method  the  linear  element the and  system  method.  test  In  employed  optimum  mesh  77  (  START  thermal d l f f u s i v i t y temperature profile boron oxide t h i c k n e s s pressure growth v e l o c i t y c r y s t a l g e o m e t r y : r , z , cone a n g l e , s e e d l e n g t h , seed l e n g t h , seed diameter. Q  •esh  INITIALIZE  [K ], T  For  each  : number o f n o d e s , e l e m e n t s , nodes w i t h constant temperature (interface), b o u n d a r y segments ( v e r t i c a l , tilted, h o r i z o n t a l ) n o d a l c o o r d i n a t e s , system topology.  :  [K ], H  0  [K ], A  element  EVALUATE  ASSEMBLE  data  :  [k ]  global matrix conditions  e  T  [  ]  without  r e g a r d i n g boundary  78  For  each  boundary  ACCOUNT FOR  convection -  CALCULATE c o r r e s p o n d i n g h e a t transfer c o e f f i c i e n t and ambient temperature  -  EVALUATE  -  For  each  node  MODIFY  SOLVE  element  [K ]  and  [K ]  [K ]  and  [K ]  H  ADD  T  with  A  H  s p e c i f i e d temperature  [K ] A  the l i n e a r Choleski's  and  [K^]  system method.  PRINT  temperatures  PLOT  temperature  to  [K  ] {g}  be r e a d  =  [K  ]  using  A  by  PFEMS  ^  contours  STOP  Figure  8.1  Flow c h a r t of the temperature f i e l d element method.  computer in the  program crystal  to calculate the using a finite  (  S T A R T ")  INPUT Poisson ratio c r y s t a l dimensions ; r , Z • e s h d a t a : number o f n o d e s , e l e m e n t s , nodes with s p e c i f i e d displacement, nodals c o o r d i n a t e s , system topology nodal temperatures  INITIALIZE  For  each  :  [ K ' ] , {F^}  element,  at each  EVALUATE  [k ] 1  (e) ij  using FORM  ASSEMBLE  node *  quintic  matrices  Global  Matrices  (e) <F' } ' i order  [k']  e  and ( F ^ }  [K']  w i t h o u t r e g a r d i n g nodes displacement  5  integration  and  e  in  element  {F^}  with  specified  80  2 ACCOUNT  for  nodes  SOLVE  the  linear  using  For  each  system  Choleski's  displacements  OBTAIN  element  CALCULATE  specified  with  at each strain  [K ]  {d }  1  1  method  stress  0  u, w  node  :  components  e  , e„, e  e  components  o  , o„,  e  P  . At each  <F }  and  P  CALCULATE  displacement  , Y  c o  pc , T PC  C  node  CALCULATE  average stress c a l c u l a t e d elements sharing the node  from a l l  CALCULATE  principal  , o  stresses  o  , 0 J.  CALCULATE  Von  Mises  stress  0 V Fl  O  81  Output Stress nodes  PRINT  PLOT  Von  components  Mises  stress  and Von M i s e s  stress  at a l l  contours  STOP  Figure  8. 2  Flow c h a r t o f t h e calculations using elements.  computer finite  program linear  f o r the s t r e s s and quadratic  82  (  /^INPUT  START  )  : element s i z e , c r y s t a l length, seed w i d t h , seed l e n g t h , cone  INITIALIZE  CALCULATE  :  :  each  first  axial  GENERATE  last  element  calculated  number o f n o d e s i n r a d i a l a n d a x i a l directions recalculate element size f o r the bulk crystal  GENERATE  For  l a s t node and LAST, L A S T E L .  angle  row  position nodes  o f nodes  at the  i n the bulk  i n the r a d i a l  interface  crystal, direction  83  2 For  each  eleient  GENERATE  bulk  CALCULATE  For  For  each  i n the bulk crystal  number  axial  crystal, topology  and d i m e n s i o n o f b l o c k s  position  i n the cone,  CALCULATE  number o f nodes  CALCULATE  element  GENERATE  nodes  each  element  GENERATE  CALCULATE  i n t h e cone  size  i n the radial  In radial  in radial  direction  direction  directions  i n t h e cone, cone  topology  number o f nodes In t h e seed  5  in radial  and a x i a l  directions  84  © For  each  axial  GENERATE  For  each  /PRINT  nodes  element  GENERATE  DETERMINE  position  i n the seed,  i n the radial  direction  i n the seed,  seed  topology  segments where c o n v e c t i o n - vertical segments - tilted segments - horizontal segments  Data  t o be r e a d b y PFEMT  occurs,  a n d PFEMS  /  STOP  Figure  8.3  Flow  chart  o f t h e mesh g e n e r a t o r c o m p u t e r  program.  85  consisted of  the  in  of  crystal.  the  and  and  The gave  unit  value.  FEM,  In  to  To for  was  nodal  of  large  solid  i n such  and  heat  transfer interface.  conditions  should  method  which  surface  temperature  liquid  solution  mesh,  the  ambient  constant  the  coarse  of  The  nodal  should  were  were  be  selected  99  %  of  the  the  nodal  nodal  Von  crystal.  Mises  in  linear  stress  10  - 4  out  in  using  such  simple  matrices  matrix  were  should  requirements, for  be  which the  is  whole  array.  effort  stress  of  and  the  and  Von  temperature  isotherm  program  components  stresses  .  the  zero.  temperatures the  that  element  memory  with  throughout  (eigenvalues  computational of  be  stiffness  in a  field  less  stiffness  computer  calculations  carried  defined  output  displacements,  i n the  tests  directly  the  values  stress  systems,  stress  temperature  stress  the  the  consisted  crystal.  contours  type  unit  positively  minimize  output  coordinates,  a  that  assembled  calculations the  at  thermoelastic  manual  ensure and  critical  The  the  theoretical  addition,  positive).  for a  nondimensional  the  symmetric  system  of  application  conditions  crystal,  crystal  mesh  fields  case  gave  meshes  of  unit  on  value.  the  crystal  in  i n the  temperature  In  the  field  conditions  were  temperature  combination  theoretical  the  unit  have  boundary  conditions  surrounding  temperature  uniform  uniform  These  medium  coefficient The  applying  plots  consisted  in Mises  of  cylindrical isostress  86  Two  other  component parallel plane the  of  and (010) p l a n e difference  (CRSS)  or  yield  values are  have  of nodal plane  the  shear  i n Figures  (specific  axis.  These  planes  and  stresses  been  8.4  and  correspond plots  CRSS  f o r the  In  which  to a (001)  values  charts  gives  of  generation and  Te-doped  comparison.  a l l cases  and  are obtained of  f o r doped  of the corresponding  and d i r e c t i o n )  perpendicular  dislocation  The f l o w  8.5.  t h e maximum  experimental  for  employed  dependent.  values  Similar  MRSS  of  for calculating i n planes  respectively.  are temperature  consists  stress  (YS). Values  GaAs  shown  written  shear  between  resolved  [CRSS(Te)]  These  of the programs the output  stress  also  a n d t h e mode  t h e maximum  component  stress.  The in  resolved  were  to the crystal  critical  of  programs  computer  programs  f o r the four  calculations  are  listed  Appendix XI.  8.1.1  Input  The  input  computer mesh  Parameters  parameters  programs  generation, 1)  crystal -  2)  depend only  f o r the c a l c u l a t i o n s on  the stage  geometric  dimensions  being  parameters  i n the  different  modelled.  are given,  they  :  radius length cone a n g l e seed l e n g t h seed r a d i u s i n t e r f a c e shape  element  size  given  by t h e d i s t a n c e  between  nodes.  For the are :  /'INPUT number o f n o d e s stress components nodal coordinates temperatures  At  each  node  -  CALCULATE f i v e components o f t h e r e s o l v e d s t r e s s e s In (010) p l a n e s (RSS)  -  OBTAIN  -  SUBSTRACT c o r r e s p o n d i n g critical stress (MRSS> - C R S S )  maximum c o m p o n e n t GET n o d e  PRINT  :  MRSS  - CRSS,  PLOT  :  MRSS  - CRSS  8.4  Flow plot  RSS  (MRSS)  resolved  »  shear  mode  contours  ^  Figure  of  shear  STOP^  c h a r t of the computer program RSS i n a v e r t i c a l (010) p l a n e  to calculate  and  INPUT  -  GENERATE  polar  At  L  each  s t r e s s components temperatures  polar  d i r e c t i o n at Z  grid  node  -  CALCULATE  t e n components s t r e s s (RSS)  -  OBTAIN  maximum c o m p o n e n t o f s t r e s s GET mode  -  SUBSTRACT  corresponding c r i t i c a l stress (MRSS - C R S S )  PRINT  PLOT  :  :  MRSS  MRSS  - CRSS,  - CRSS  8.4  Flow plot  of the resolved  shear  (NRSS) ;  resolved  shear  mode  contours  (  Figure  in radial  STOP  )  chart of the computer RSS i n a (001) p l a n e  program  to c a l c u l a t e  and  89  The  input  for  the  temperature  configuration,  the  nodes  heat  is  transferred  data  includes  profiles at  the  The  in  gas  both  h  =  The  media.  h  r  19 0 following  (2 . 27  is  temperature  layer  transfer  equation  e  The  the and  these  the  height  heat  transfer  by  The  and  calculated to  where  physical  temperature coefficient  in  radiation  element  segments  transfer  relative  coefficient  the  for  Stefan-Bo1tzman  the  program.  the  thermal  and  convection  radiation  was  derived  by  convection  x  10  1 1  /  total  doping  rem" ]  =  transfer assuming wall  - T  (  of  GaAs  and  1  /  coefficient heat  into  a  is  _JL_ K.  is  a  the  function  fluid  4  -)  h"  p  ^  /K^J /  1  /  [ p  2  G  ^  2  1  a  to  the  gas  transferred  where  5 4 8  to  3  1  C  .  leading  of  level.  a vertical  T 1  K ) £ T t a  emittance  calculated from  equation  :  c o n v e c t i v e heat  was  h  the  and  c  from  =  r  h  oxide  coefficient  and  nodal  atmosphere.  boric  is  the  191 '  where  +  surrounding  From  heat  is  interface  temperature  includes  heat  Jordan  the  transfer  conductivity  the  pressure,  corresponding  heat  h  to  at  program  2  and  i s given  and  B^Og  by  free  by  191  90  h  tooo  1100  1200  1300  TEMPERATURE  Figure  8.6  rod  1400 (  K)  Estimated radiative and convection heat transfer c o e f f i c i e n t s f o r G a A s / B 0^ ( 1 ) , He ( g ) , N (g) and A (g) as a function or ambient temperature. The numerical labels are the product of the c o n c e n t r a t i o n X c r y s t a l d i a m e t e r i n u n i t s o f cm g  carrie  Figure  8.7  T o t a l h e a t t r a n s f e r c o e f f i c i e n t h, i n 2°3 * £ as a f u n c t i o n o f t h e a m b i e n t t e m p e r a t u r e T (1) T o t a l heat t r a n s f e r c o e f f i c i e n t in 2°3' l heat t r a n s f e r c o e f f i c i e n t i n argon p r e s s u r i z e d a t 30 a t m . B  B  a n (  T  o  t  a  a  r  o  n  91  K  =  ambient  thermal  =  ambient  density  =  coefficient  Cp  =  heat  u  =  viscosity  p  =  pressure  1  =  height  p a  for  a  a  a g  a  liquid  The  of  diffusivity  thermal  volume  expansion  capacity  of  medium  numerical  the  p  fluid  i s taken  evaluation  column  as  p  =  of  the  1.  heat  transfer  coefficients  191 was  done  of  the  by  Jordan  physical  pressure  of  1  different  media.  oxide  argon  the  and  program  functions  The at  30  which  f i t  and  and  same  temperatures nodal  transfer  nodal be  values  the  Poisson most  Figure  in  Other  physical  is  8.6  Figure  are  include  assumed  a  gas  for  the  for  boron  8.7  In  evaluated  the  parameters to  values  for  coefficients  shown  and  dependent  results  coefficients  be  by  effect  of  are  the  temperature  velocity.  configuration used  temperatures the  of  growth  are  data  which  in  transfer  atmospheres  the  The  shown  heat  pressure.  must  addition  are  total  temperature  involved.  atmosphere  diffusivity  The  using  quantities  heat  independent,  The  al.  the  temperature thermal  et  for are  ratio  important  employed  the the  and  in  thermal input  Young  parameters  the  stress  for  listed  of  calculations.  this  modulus are  calculation  program.  are  given.  in Table  In The 8.1.  92  Table  8.1  Values of p h y s i c a l calculations  parameters used  Melting  point  1238°C  Thermal  conductivity  0.08  Thermal  diffusivity  0.04, ± 0.01 (cm / s e c )  Watts/cm  Thermal expansion coefficient  1.0  Young  1 . 2 X  modulus  Poisson  a E /  ratio  X  the  MRSS  10  2 . 86 ± MPa  yield  10~  5  the  K  (800  (°K  C  -  1238  C)  *)  Pa  1 1  0 . 29  (l-2v)  The  In  and  were  critical  stresses  measured  by  °K  0.1  (800  C  1238  C)  _ 1  employed  Swaminathan  f o r comparison and  Copley  211  with and  16 3 Mildviskii  et  temperatures this  point  assumed  as  a l . well  ; the  an  respectively. below  inverse  the  The  melting  point  exponential  experimental  results  yield  stress are  extrapolated  temperature suggest.  The  measured  dependence  CRSS  for  at to was  undoped  10  o.)  07 1  /2V  Figure  8.8  i  /too  1  1001  as 1  KO  It  .L t00  Temperature dependence of the critical stress for dislocation generatjgn ig GaAs (1) Te-doped m a t e r i a l , n_= 2 10 cm ; (2) T e - d o p e d material, X cm ; (3) Zn-doped m a t e r i a l , p = 9 X " 1 8 ( 4 ) u n d o p e d m aterial. 10 cm 7  4°"  93  and  doped  8.8.  It  one  order  8.1.2  GaAs  can  be  of  the  the  roots  evaluated The between  were  20  to  40  slow  convergency  8.2  Comparison Solutions  8.2.1  in  Temperature  Field  Section  heat 0.6 a 20  *,  are  pressurized mm  and  with  the  the  using  in  VIC-20 A  BASIC  value,  graphic  J ,  ,  Q  series  because  of  were  213  out the  Predictions  obtained  with  using typical  Analytical  analytical solutions  for  the  problem  performed  for  two  coefficient range  were  h.  mm.  The  values  applicable  atmosphere 40  first  functions  carried  of  1^  and  Q  finite  height  with  . The I  the  was  necessary  language  solutions  methods  for  for  microcomputer  the  argon  about  Solutions  using  the  is  analytical solutions  function  was  Figure  material.  programs a  in  material  undoped  The  Fourier  Calculations  in  of  shown  series.  solutions  7.3.  transfer cm  this  doped  interpolations  This  are  Analytical  in  Bessel  the  Model  compared  the  KRam.  The  terms.  for  for  run  obtained  of  Numerical were  8  of  of  than  point  conditions,  polynomial  summation  CRSS  of  were  to  used.  using  the  evaluation and  were  melting  larger  Programs  (7.45)  the  Evaluation  memory  Equation five  that  fields  written.  expanded  noted  numerical  different  were  to  magnitude  Numerical  For  close  The  of  to  values 0.3  crystal  being  a  are  The  crystal  shown  in  method  described  selected,  3.04MPa.  results  element  of  cm  1  grown  radius  Figures  the and in was 8.9  94  Figure  8.9  A comparison of finite element c a l c u l a t e d temperature curves for h  =  and 0.3  analytical cm  95  analytical and = 0.6 cm F igure  8.10  96  Figure  8.11  Mesh e m p l o y e d i n the c a l c u l a t i o n s of the temperature f i e l d s shown i n F i g . 8.9 a n d 8 . 1 0 . Number o f n o d e s = 45. Number o f e l e m e n t s = 64.  97  and  8.10  finite than  from  which  element the  method  were  of  Further  i s evident gives  temperatures  elements number  It  was  in  45.  refinement  of  the  temperatures  calculated  employed  nodes  that  the  The  the  agreement  which  slightly  analytically.  calculation  mesh mesh  are  used did  and  is  not  i s good.  A  shown  lower  total  the  The  of  64  corresponding  in  Figure  appreciably  8.11.  change  the  results.  8.2.2  Stress  Finite crystal linear  Fields  element  for  solutions  radial  elements.  A  and  into  nodes.  stresses  The  computed element  nodal  element 8.12.  by  radial  calculated  mesh  by  of  were  the  256  and  finite  element plotted  the  displacements.  the well  using  finite below  also  two the  element the  shown  average  other  three.  This  using  40  of  mm  was  of  153  a)  the  temperature  radial  for  nodal  function  in  the  of  the  computed large  nodal  Figure  determined strain  stresses  temperature.  average  r / r ^ in  plane  and  stresses  and  stresses  both  solutions  using  basis  the  radial  element  fields  temperatures.  for  The  analytical  method  a  cylindrical  consists  the  uniform  method as  a  length  which  corresponding  axisymmetric  solutions  a  fields,  are  the  b)  and  on  temperature  are  mm,  elements  nodal  comparison  for  20  calculated  analytically  coincide  radius  for  temperature  a v e r a g i n g the  temperatures For  of  temperatures  derived  For  a  obtained  axisymmetric  crystal  discretized  were  effectively  finite The  element  results  temperatures  difference  and  is  for is  attributed  98  -02  CO I  Q_  -0 4 -  if UJ - 0 6  -  -0 8 -  r/r  r  Figure  8.12  C a l c u l a t e d r a d i a l s t r e s s e s as a r a d i a l temperature f i e l d s (1) averaged element (2) F i n i t e temperatures (3) Analytical Analytical-axisymmetric.  f u n c t i o n of r / r for F i n i t e element with element with nodal plane strain (4)  99  Figure  8.13  C a l c u l a t e d a z i m u t h a l s t r e s s e s as a f u n c t i o n o f r / r for r a d i a l temperature f i e l d s (1) F i n i t e elemen? with averaged element temperatures (2) Finite element with nodal temperatures (3) Analyticalplane s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c .  100  i  -0-6  0  02  1  1  04  06  r  08  10  r/r.  Figure  8.14  C a l c u l a t e d a x i a l s t r e s s e s as a f u n c t i o n o f r / r for r a d i a l temperature f i e l d s (1) F i n i t e e l e m e n t with averaged element temperatures (2) F i n i t e element with nodal temperatures (3) Ana 1 y t i c a l - p 1 a n e strain (4) A n a l y t i c a l - a x i s y m r a e t r i c .  101  to  the Incompatibility  comparison nodal  with  while  and  the linear  within  the  element,  respectively. the  radial  solution  different The  curve  element given  deviates  from  compressive  To stress  from  test  i n Figure  axial  nodal  than  is  and  The  other  the r a d i a l  the  the  temperature  other  Q  8.13  and  a  element  strain small  curve  is  plane  giving  maximum  curve  amount  as  not  nodal  a  8.14  with have  a  solutions.  element  curve  to  analytical  i n Figure  finite  8.14  similar  giving  analytical  two  of r / r f o r  and a z i m u t h a l  axisymmetric  The  function  plane  two  the  field.  strain  curves  between  previously. the  three.  stress  within  the f i n i t e  thermal  nodal  and i s c o n s i s t e n t  Figures  with  strain  the  8.13 a r e g e n e r a l l y  8.10  the  as a  the  element  a constant  constant  constant  in  the  When  in  I f the  both  within  the displacement  shown  the other  agreement  also  introducing  configuration  temperature  reasons  below  r / r ^ . The  best  finite  of  assumed  stresses,  in Figure  deviates  increasing  with  are  averaging,  linearly  strain  element.  inconsistent.  is  and a x i a l  method  well  is  strain  The r e s u l t s  vary  and  associated  stresses  temperature falling  which  solution  without  formulation gives  averaged  azimuthal  same  strain  of i n i t i a l  i n the l i n e a r  used,  displacement  the i n i t i a l  The the  initial  are  the s t r a i n  field  are  element,  temperatures  with  the s t r a i n  temperatures  temperature  i n the approximation  average  before.  valid  The  f o r the  strain tensile  values and  stresses.  the  convergence  calculations  were  made  of  the  with  finite  refined  element meshes  procedure, for a  radial  Figure  8.15  Four s t e p s i n the gency of the f i n i t NE = 16 ( b ) NN = 153, NE = 256 l i n e a r elements.  mesh r e f i n e m e n t u s e d t o a n a l y s e t h e conver e element s t r e s s c a l c u l a t i o n s . ( a ) NN = 15 45, NE = 16 ( c ) NN = 4 5 , NE = 6 4 . ( d ) NN (b) Q u a d r a t i c e l e m e n t s . ( a ) , ( c ) and (d ,_, o to  103  0  NODES  -02  L  45  64  •  1 Q  45  16  A  L L  153 15  256 16  L  45  64  Q  45  16  L  153  256  -03 2  -0-4  CM I  ELEMENTS  •  A O  •o. - 0 - 5  2  -0-6 -07  -0-8 -0-9'  10  Figure  8.16  C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r / r f o r different numbers of nodes and s i z e s of element (1) F i n i t e e l e m e n t w i t h a v e r a g e d e l e m e n t t e m p e r a t u r e (2) F i n i t e element with nodal temperatures. L = L i n e a r element, Q = Q u a d r a t i c element.  104  temperature for  a  the  mesh  cylinder  stress node  Calculations  of  half  refinement  temperatures  have  no  cases.  that  and  large  nodal  element  and  described  It  does  stresses.  elements  are  seen  Figure  in  quadratic coarser  finite  of  elements  the  and  a  using  that  number  the  of  quadratic the  elements in  associated  fewer  as  element in  and  the  larger  formulation,  improved is  with  strains  difference  nodes  both  between  quadratic  element  displacement  and  between  the  results  difference  i s not  in  individual  stresses  large  of  The  nodes  significant  hence  of  element  radius  As  20  256  differs has  curve  before,  markedly a  higher  analytical  field  corresponds  temperature  axisymmetric  and  thermal  field  were  average  curve  of  steps  as  accuracy  of  the  compromised  by  the  stress  fields  for  Figure  8.17.  The  grid.  temperature  value.  same  8.16,  the  of r / r ^  results  temperatures.  calculated  evident  the in  8.16  temperature  to  For  The  incompatibility  also  function different  8.15.  number  that  lead  involved  axisymmetric  crystal  the  a  The  Figure  the  the  interpolation  The an  on  element  is  not  computed  in  in  as  length.  element  show  with  above.  formulation  also  made  in Figure  shown  effect  but  in  average  averaged  size  shown  changes  results  were  radius  are  significant The  a  are  calculations  indicate  the  field.  mm  and  and  for the  from  shown  an  h  40  in  value  mm.  nearly Q  finite  element  others.  compressive  0.5,  stress  The  0.3  number The  coincides  above  of  The  respectively.  r / r  the  to  length  153  curve  are  radial  cm of  the  deviating  than  the  and  analytical below  plane  others.  a  element  temperature  analytical  for  nodes  finite  with  nodal  *  this curve strain  105  Figure  8.17  C a l c u l a t e d r a d i a l s t r e s s e s as a f u n c t i o n o f r / r for axisymmetric thermal fields (1) F i n i t e element with averaged element temperatures (2) Finite element with nodal temperatures (3) Analyticalplane s t r a i n (4) A n a l y t i c a l - a x i s y m m e t r i c .  106  The finite slow  difference element  first  the  stress  field  involving  of  terms  rapidly radial the  with  complex  of  the  two  solution  to  Recently  solutions  semi-infinite  are  done  isothermal for  effect  (the  unit  is of  order  to  displacement  only  of  summations  very  serious  when  simplicity  of  as  to  in  field.  the  have J  opposed the  The a  case  been  of  inclusion  more  Bessel  were  diverge  the  obtain  values.  differential  equations  including  solution  with  not  expanded  temperature  of  linearly  function),  is  the  use  in  functions  because  in  the  values  increases  Bessel  which  and  and  to  precise  terms  used  similar  cylinders  two  This  function  was  less  oscillatory  1^  dimensional  functions the  the  with  analytical  in  curve  attributed  combined  argument  i s expanded  z-dependent  axisymmetric I  and  fields  which  an  The  is  the  last  arguments.  temperature  function  is  the  terms.  these  series in  series  axisymmetric  curve  results  averaging 40  number  Fourier  which  about  analytical  function  this  by  the  temperature  the  Bessel  in  calculated  the  of  order  Convergence  a  average  convergence  the  between  general  equation.  obtained  in  functions  in  16 8 a  less  general  The  results  temperature which  are  element  more  good  shows the  behaved  Figure the  8.17  plane  double  using  were  where  for  in  than  method  crystal  better  fields  differences  result  but  usually  also  strain  the  the  solution show  obtained  axisymmetric close  c o n d i t i o n s d e v i a t e markedly that  the  analysis  plane of  strain  thermally  for  approximation  stresses  observed  that  gives with  stresses  the  finite  approximations.  Larger  to  longer  from  the  ends  plane  approximation induced  axisymmetric  stresses.  of  strain.  is  not  It  This always  should  be  107  noted  that  l i t e r a t u r e  In  this 2  1  "  4  2  1  7  the  has  i t has  finite  been  and  the  assumption.  The  analytical  movement finite  does in  this  element  case.  when  from  reported  model  may  due  in  that  the  agreement element  axisymmetric  the  plane  restriction  are  with  that  strain of  the  comparable  the  axial linear to  plane  the  strain  applies.  Predictions  compared  the  concluded  stresses  condition  the  good  average  with  with  be  and  is  using  to  thus  distribution  is  there  solution  well  gives  Model  temperature  the  It  that  Comparison of Measurements  The  f i t as  that  solution  approximation  approximation  been  analysis  analytical  analysis  axisymmetric  8.3  not  shown  element  temperatures  assumption  not  .  summary,  between  result  with  in a  to  Temperature  growing  the  crystal  results  calculated  of  temperature  218 measurements temperature  made  crystal growth  to was  period  a  measurements  thermocouples attached  in  shown the  in  seed  grown. of  LEC  GaAs  were  made  Figure  holder  and  crystal,  the  The  moved were  giving  crystal  with  8.18.  Temperatures  the  Melbourn  up  array  of  with  the  seed  for  results  were as  the  shown  The four  thermocouples  recorded  the  grower.  the  entire  in  Figure  8.19. To  calculate  the  temperature  distribution  with  the  model,  2 the  thermal  transfer were  diffusivity  coefficients  determined  from  was  between the  taken the  values  as  crystal  given  0.04  cm  and  the  in section  /s. argon 8.1.  The and  heat BO  108  Figure  8.18  P o s i t i o n o f t h e r m o c o u p l e s i n GaAs c r y s t a l . B = B o r i c o x i d e l a y e r , a r g o n p r e s s u r e 3.04 MPa. Reference 218.  109  The shown  ambient  in  Figure  measurements different the  temperature 8.19  growth  On  basis  the  temperature  segment  temperature the  axis A  valid with  at the  Figure  and  15°C.  calculated  measured  curves  temperature point  length.  grows.  the  this  at with  temperature  after  lengths and  being  are  at  crystal.  The  change  at  points  with  time  these mm),  the by  increases  A,  B,  the  longer  C  the with  only  and  is  D  in the  close  measured that  ; and the  crystals  curve  temperature  is  between  points the  8.20.  coincides  indicating  calculated  that  Figure  Agreement  coincide, For  in  crystal  temperatures  marked  (55.0  the  temperature  lengths. at  along  shown  measured  measured  difference  will  ambient  coefficient  growing  calculated  the  i.e.  the  temperatures  crystal  below  measured  melt.  independent.  are  The  crystal  distance  transfer  crystal  temperatures  effectively time  the  crystal  short  dependence,  decreases  crystal  four  the  is  to  crystal,  temperature  the  the  heat  of  measured crystal  measured  the  the  temperature,  curves  curve  from  where  the  For  that  the  on  on  assumption,  calculated  point  for  point  assumed  is  within  and  the  measured  calculated  measured  of  based  time  surface  is pulled  the  8.20  within  the  different  comparison  is  converting  determine  distribution  four  fixed  this  to  calculated  at  4),  along  independent.  along  crystal  The  It  of  i s used  a  growth,  velocity. i s time  as  to  during  distribution  each  (curve  adjacent  times  distribution  the  indicating at  thermocouple increasing  a  given as  the  crystal  110  •20  Figure  8.19  0 20 4 0 60 80 100 120 140 160 Position Relative To Interface (mm)  T e m p e r a t u r e s m e a s u r e d w i t h t h e r m o c o u p l e s 2, 3 a n d 4 in Figure 8.18 as a function of the r e l a t i v e p o s i t i o n of the thermocouples with the interface. Reference 218.  Ill  —  —  i  1  1  1  1  1  1  1  1  1  r  1300  Position  Figure  8.20  Relative  To  Interface  (mm)  Measured and calculated temperatures c r y s t a l a x i s at f o u r c r y s t a l l e n g t h s .  along  the  112  Figure  8.21  Measured outside lengths. shown.  and c a l c u l a t e d t e m p e r a t u r e s a d j a c e n t to the crystal at four c r y s t a l surface of the The measured ambi ent temperature is also  113  81  1  l  1  0  Figure  8.22  1  1  1  1  20  1  1  1  1  40 Position Relative  1  i  1  i  1  •  60 80 To Interface (mm)  Measured and c a l c u l a t e d axial a l o n g the c r y s t a l a x i s at f o u r  1  •  1  I  100  r  L  temperature g r a d i e n t s crystal lengths.  114  The both in  temperature  calculated Figure  included curve  i n the was  temperature at  the  The  and  curve  model  are  the  below  the  1 and  presence gives  these  the  melt  lower  a  For  longer  the  calculated  length. values  At are  scatter  axial  in  Figure  used  is  shown  is  also  interface,  as  the  ambient  i s assumed  shows  flat  and  calculated  there  is  and  reasonable  to  where  than  most  The  a  the  crystal,  are  growth  by  compared  to  measured  gradients  gradients  gradients crystal.  i s shown flat  measured  the  during  The  near  may  In  later  addition, in this  measurements.  about  increasing  comparison values  are  but  is valid within  the  correspond  interface in this  gradients  by  length.  results  8.22.  values  the  of  crystal  measured  difference  calculated  experimental  over  the  as  measured  temperatures  measured  measured the  the  calculated  partly  gradients  the  the  the  than  8.18.  interface,  points  D  values  also  the  the  was  increasing  was  points  values,  above of  the  the  crystals  the  lower  in Figure  and  and  crystal,  lengths  to  interface  are  below  convex  axial  C  the  distribution  Comparing  calculated  gradients  At  of  with  2  (b)  of  values.  gradient  significantly  to  two  crystal  adjacent  the  B,  i n c r e a s i n g with  calculated  partly  A,  values  As  axial  interface.  since  at points  edge  temperature  temperature.  25°C.  thermocouples  the  that  the  difference  are  Note  calculated  The  the  figure.  between  than  values the  in  the  f o r four  ambient  measured  values  agreement  measured,  near  The  freezing  measured  less  and  8.21.  (a)  distribution  10  the  work, region.  %  below  with  crystal  the  measured  the  range  of  115  In and  summary,  measured  the  valid  temperatures  temperatures  predicted  temperatures  i n the  8.4  time  calculate  dependent  program is  is  shown  by  comparison  i n good  the  Model  the  of  agreement.  model  are  for Cooling  temperature  solutions,  shown  in  are  of  the  calculated  Accordingly,  indicative  of  the  the  real  crystal.  The T e m p e r a t u r e Convergency  To  points  in  Figure  a  Appendix 8.23.  - Programming  fields  based  FORTRAN  program  XI.  flow  The  The  value  the  was  chart  X  of  on  were  and  analytical  written. of  this  This  program  calculated  using  219 cubic  interpolation  used.  The  Newton were  -  Y  -  v  l  a  Raphson  e  were  s  method  tables  ,  obtained to  the  from  find  the  first  six  Equation  roots.  roots  (7.41b)  The  first  were  using  30  a  values  used. With  converged part  of  these  six  to  %  the  98  exact  A shown  radial  value.  converged was  to  part  With 98  %  therefore  30 of  of Y  the  temperature  values  the  i t s final  calculated  axial  value. as  96  The *  of  solution.  Figure  the  length  the  crystal  axial  the  i t s final  temperature  typical in  of  X values,  temperature  non-dimensional the  u  from  is  temperature  field  calculated  8.24  radius  of  110.0  is  a  high  what  i s observed  The  mm.  Due  shown.  temperature  Also  a.  Close  gradient  radial  gradient  during  to  symmetry,  to  the  changes is  growth.  the  crystal only  interface from  usually In  during  view  the or  positive observed  of  this  and  cooling  is  i s 27.5  mm  right  half  tail to as  end  and of the  negative. opposed  the  fact  to that  116  (  START  /INPUT  -  For  y eigenvalues 3 eigenvalues c r y s t a l r a d i u s and l e n g t h heat t r a n s f e r c o e f f i c i e n t , ambient temperature i n i t i a l a x i a l temperature  each 3 : CALCULATE  For  coefficient  (3)  coefficient  C (y)  each Y : CALCULATE  For  h  each  radial  EVALUATE  position series  : i n Bessel  5  functions  117  For  each  axial  EVALUATE  For  each  point  CALCULATE  / PRINT  /  j  PLOT  nodal  1  position series  i n the  in Fourier  functions  grid  temperature  temperatures  /  /  /  isotherms  STOP  Figure  8.23  Flow c h a r t of the computer program f o r the numerical evaluation of the analytical temperature fields during cooling of the c r y s t a l .  118  b  a Figure  8.24  e  (a) T y p i c a l temperature field obtained during cooling units are 10 °C. (b) and ( c ) Von Mises stress c o n t o u r s (MPa) for the temperature field given in (a). ( b ) NN = 451, NE = 800. ( c ) NN =  1105,  NE = 2048.  119  for  the s t r e s s  mesh a  refinement  given  mesh  The  results  the  Von  part  of  that  values  coarser  field  figure,  t h e same 10 The  quadratically  mesh  was  of  8.24  1105  less  done  are used  the  using  a  nodes  and  pattern  t h e same 2048  i n the coarse  t h e number  shows  field  in 451  stress  field  elements.  It is  mesh  with  stress  following  involved,  of nodes.  b  containing  i s obtained  effort  8.24  temperature mesh  i ff o r  representative.  b and c. F i g u r e by  another  to determine were  c shows  computing  with  selected.  8.24  stress *  was  stresses  Induced  Figure  exactly  a r e about  This  calculated  a mesh  tendency.  increased  i n Figure  elements.  temperatures  calculated  employing  which  expected  the  stress  the  average  performed.  a r e shown  a n d 800  obtained  was  size  Mises  (a)  nodes  seen  calculation  an  however  Accordingly  the  120  CHAPTER  RESULTS  The used  to  mathematical study  dislocations The  variables  the  following  1.  2.  effect  during  angle  following  cooling  to  crystal  the  radius  variables  by  previous  variables to  geometry  (CA), given  the  in  ambient during  angle  chapters  is  generating  temperature.  growth  include  between  the  cone  horizontal (CL),  solid-liquid  given  interface  by and  the the  distance  start  of  between  the  cone  (R),  shape  associated with  the  growth  process  include  the  :  5.  Growth  6.  Boron  7.  Thermal  8.  Gas  9.  Gas  The  and  length  Interface  The  growth  Crystal  4.  i n the  different  and  Crystal  presented the  surface  3.  ANALYSIS  of  related  Cone  the  the  the  model  AND  9  velocity oxide  (V)  thickness  gradients  pressure  (B)  (argon,  AG  and  boron  oxide,  BG)  (GP)  composition  variables  following :  associated  with  the  cooling  process  include  121  10.  The  nature  immersed.  of  This  11.  Temperature  12.  Thermal  the  may  media  in  be b o r o n  oxide  of the cooling  conditions  in  which  the  or argon  crystal  is  gas  media  the  crystal  at  the start  of  cooling.  The fields  9.1  Cone  ; ;  The  cone  degrees.  The  coincident  plane  High  a r e now  on  the calculated  stress  considered.  54.7°  10 mm  ;  BG , 100 ° C / c m  ;  are  7.1,  corresponds  30,  to  45,  the  54.7  and  cone  65  surface  t h e (111) p l a n e .  Von  five  The  stress  angles  Von M i s e s  t h e cone  levels  Mises  cone  T h e maximum  axis.  B,  considered  angle  where  ;  Assumed  50°C/cm.  angles  with  stress  crystal seed .  AG,  f o r the  shoulder  Parameters  calculated  9.1(a-e).  density  C L , 10 mm  30 atm  The  variables  Angle  and Growth  20 mm  AP,  o f t h e above  and d i s l o c a t i o n  Crystal R,  effect  reaches  occur lowest  along  distribution  considered Stress  (MVMS)  the f u l l both  Von M i s e s  is  stresses  shown  levels  radius  the  on  cone  of  a  vertical  in  Figure  occur  at the  the  crystal.  surface  and t h e  (LVMS)  a r e below the  122  123  Figure  9.1  Von Mises Stress c o n t o u r s (MPa) in vertical planes f o r f i v e cone a n g l e s . (a) 7.1°, (b) 3 0 ° , ( c ) 45 , (d) 5 4 . 7 ° , (e) 6 5 ° . Cone s u r f a c e i n (d) coincides w i t h a (111) plane. C r y s t a l r a d i u s , 20 mm ; crystal length, 10 mm ; 2°3 thickness, 10 mm ; 2°3 g r a d i e n t , 100 C/cm ; argon pressure, 30 atm. ; argon g r a d i e n t , 50 C/cm. B  B  124  \Table  9.1  Effect  o f Cone  Thermal  Angle  on T h e r m a l  Angle  (°)|  ° C / c m | 1 % V™ rOue* _ MVMS I AMRSS I AMRSS/ Axial | MPa MPa MVMS  The  6.7 4.1 2.7 3.2 2.5  largest  crystal  with  appreciably and  when  reaches  increases  between  VMS  52.4°  to account  two  sources  field  radial  axisymmetric additional  determined  thermal  body stresses  amount  stress  the  does  7 1 5 6 3  occurs  from  cone  angle  o f cone  of  i n the  combined  drops 30°,  57.4°  then  i s observed  angle.  o f t h e cone  of thermal  have  .47 . 46 . 48 . 4 . 45  30° and 4 5 ° .  behaviour  in  0 0 0 2 0  7.1° t o  angle  are considered  stresses  develop  9.1,  between  by t h e t h e r m a l  not  8. 3 . 2 . 2 . 2 .  7 . 1 ° . T h e MVMS  geometrical  gradients  large  may  of  f o rthe effect  and  s l  7 2 1 0  i s increased  for a  i s the origin  produce  angle  as a f u n c t i o n  thermal  fields  i s roughly  gradients  of  angle  i n Table  and 65° . S i m i l a r  In  order  cone  minimum  i s considered  temperature  large  a  shown  18.4 6 . 5 . 1 . 5 .  by a s m a l l e r  t h e LVMS  temperature  level  as  t h e cone  further  MVMS  when  55 50 54 56 55  the sharpest  decreases  The  MVMS,  4  1  Radial  7.1 30 . 45. 54.7 65 .  Fields  Gradients, :  Cone  and S t r e s s  on t h e  namely  constraints. stresses.  The  the The  stress  gradients. Generally, with  large  cylinders.  constant  axial If  the  cross-section,  due t o g e o m e t r i c a l  constraints.  125  For  the crystal  are  calculated  from  as  in  the  respectively calculated which Table  from  axis  Figure  shown  about  Table  gradients  between mm  9.1(e)  the  above,  .  The  9.1(e).  gradients  are calculated interface  labelled  radial  differences  i n Figure  A  and a and  gradients  between  point  The r e s u l t s  gradients  t o t h e cone  angle,  about  B  are  B and C  are given i n  The  sharply  from  t h e MVMS  minimum  6.7°C/cm  a t 54.7°  values  MVMS  geometrical  constraints  the  a t t h e cone,  crystal cone  angle,  t h e cone  axisymmetric maximum  stress values  diagonal  stress  the  i s not zero  distribution  as w e l l  with  tensor. because  components.  of  stress  components  30°,  again  65°.  at  to the  will  two  a  For a  of  t h e VMS large  a  third  the will  non-zero  conditions, have  of  prevail  components  will  giving  i n radial  i n radius  stress  thermal  level  i n MVMS  gradient.  conditions  are only  F o r t h e same  4.2°C/cm.  attributed  and shear  such  there  a similar  stress  to plain  *  7.1° and  f o r by t h e c h a n g e i s  3  the radial  9.1, t h e d r o p  as t h e r a d i a l  axial  Under  drops  t o be  about  of  the v a r i a t i o n s  similar  zero  hand  between  i n Table  from  by  gradient  and then  f o r 54.7°  arising  conditions  surface,  observed  varying  t h e mean  7 . 1 ° a n d 3 0 ° c a n be a c c o u n t e d  gradient.  are  o f 5 3 ° c / c m . On t h e o t h e r  at 45°, r i s e s  Considering  cone  axial  appreciably  drops  further  between  the  gradient  vary  gradient  drops  9.1  insensitive  t h e mean  gradients  have  temperature  9.1.  relatively  at  here  axial  10  the temperature  are also  From  zero  The  differences  crystal  in  considered  follows.  the temperature  point  The  geometry  when  different non-zero  126  principal  stress.  components observed than  the  decreases VMS  54.7°  Under  i t was under  From  cone  angle.  intermediate a cone  The  MRSS the  angle  Figure  angle  (AMRSS) stress  shoulder  as  angles  t h e AMRSS  VMS.  i s expected  In t h i s  case  For  cone  larger  analysis.  larger  stresses  than  are  such  In  such  apply.  cone  are  angles  angle.  to increase  angles.  the  larger  conditions.  cone  in  the  VMS  that  t h e VMS  At  with  high  cone  increasing  should  minimum  i s  exist  i s  at  observed  54.7°.  of the resolved  f o r the  9.2(a-e).  five  The  by  decreases angles  of  (MRSS) i n  considered  dependence  which  crystal.  o f t h e MRSS  T h e AMRSS  9.1  similar  45°  and l a r g e r  occurs  T h e AMRSS  column  rapidly  are on  maximum  i s r e p r e s e n t a t i v e of  distributions.  i n Table  is  distributions  considering the absolute  i n t h e VMS  is listed  stress  angles  stress  The  i n the crystal  i n t h e whole  shear  cone  MRSS  distributions.  i s shown  considered  i n the  strain  that  low  minimum  developed  angles  at  increasing  i s determined  level  plane  a  of  angles  the constraints  axisymmetric  with  Therefore  included  chapter  discussion,  t h e VMS  t o t h e VMS  cone  general  F o r cone  stress  giving  are e f f e c t i v e l y  of  i n the last  decrease  be  principal  increases,  angle.  Therefore  those  the  angle  the c r y s t a l s  to  (010) planes  in  similar the  shown  cone  cone  should  maximum c o m p o n e n t  vertical shown  closer  however,  with  between  cone  crystals.  t h e more  to  angles,  for  angle  t h e above  expected  the  factor  conditions  conditions  than  as  another  these  cases  difference  dependence  low cone  that  The  5.  to that the  at the for  F o r low shown  AMRSS  the  cone  for  the  remains  127  128  Figure  9.2  Maximum r e s o l v e d s h e a r s t r e s s (MRSS) c o n t o u r s i n MPa, in vertical (010) p l a n e s for five cone angles, (a) 7 . 1 ° , (b) 3 0 ° , (c) 45°, (d) 54.7°, (e) 6 5 ° . Cone surface i n (d) c o i n c i d e s with a (111) plane. C o n d i t i o n s a r e t h e same a s i n F i g u r e 9.1 .  129  essentially angles  constant.  supports  entirely  At large  difference  AMRSS/MVMS is  less  that  c a n be  half  t h e AMRSS  54:7°  cone  traction definition  The  condition  tensor  zero.  In  components crystal to  a  In  this  (111) plane,  t h e RSS  <110>  directions.  VMS the  results  show  angle  results  o f no  ofthe  of stresses  a  onto  stress  parallel  to the  surface  parallel  t o be  these  hand,  at  planes  maxima in  the  i s an i n d i c a t i o n o f  and t h e r e f o r e  does n o t  of the material.  of using  generating concluded  to reduce  the  t o be  l i e on t h e ( 1 1 1 ) p l a n e .  materials  properties  surface  level,  are expected  the advantage  i n order  at  the projection  presenting  i t may be m i s t a k e n l y  i s beneficial  such  i n the  direction  components  in isotropic  f o r the analysis VMS  a  T h e VMS, o n t h e o t h e r  stress  occurs  difference  stress  are projections  the crysta 11ographic  These  angle  components  they  the  t h e AMRSS  assumption  to the crystal  given in  the stress  by d e f i n i t i o n  include  cone  where  The s i n g u l a r i t y  the  requires  normal  for a  For a  6  t h e VMS.  54.7°  singularity  t h e model and  from  components.  accommodate  surface.  shear  surface  cone  on g e o m e t r i c a l  column  except  low  are dependent  than  t h e MVMS.  to  o f no t r a c t i o n  since  the  double  a n d RSS  case,  must  case  than  a  at  differs  9.1  angles  At 54.7°  on a d i r e c t i o n  this  rather  t h e MRSS  F o r a l l cone  crystal  the stresses  i n Table  i s attributed  the  i n t h e AMRSS  that  angles,  seen  i s more  o f t h e VMS  stress  drop  changes  t h e MVMS.  angle  at  field  cone  i s listed.  than  large  the conclusion  on t h e t h e r m a l  constraints. The  The  t h e MRSS  over the  dislocations. that  dislocation  a 54.7°  From cone  generation. A  130  different stress  conclusion  generating  MRSS  show  cone  angle.  stress  that  i s no  given  by  densities  present  by  such  the contrary,  levels  The  obtained  dislocations  there  On  dislocation  c a n be  using  glide.  VMS  i s double  The  beneficial  i f growth  the  t h e RSS  effect  i sthe  using  using  a  the 54.7°  a r e such  changed,  at this  calculations  results  conditions  are  which  the  effect  that on  angle.  differ  significantly  from  the  13 7 calculations the  MRSS  shear For is  i s used,  stress  similar  and  levels  components vectorial  or  of  The and  c a n be  resolved  dislocation  (a)  the y i e l d t h e CRSS  Bochkareev  1 5 0  1 6 3  i s  since  i s also  associated  times VMS  with  from  generation  substracted  [CRSS  9.2(c),  larger  levels.  9.3 but  than  the  Adding  the  the concept  Moreover  of a  cannot  be  t h e TRSS  an u n l i m i t e d  number  inconsistent.  i s done  i s substracted  values.  i n Figure  the magnitude  for yield  This  case  resolved  RSS  i n Figure  dislocation  by s u b s t r a c t i n g  stress  for dislocation '  the  shown  o f components.  which  generation.  stress  shown  by c o n s i d e r i n g  determined  shear  of the twelve  i s i n c o n s i s t e n t with  addition  stress  the present  use the t o t a l  a r e seven  quantity  systems  In  distribution  than  increased  effective  for  (b)  linear  slip  movement  critical  t h e TRSS  indefinitely  secondary  larger  .  et a l .  i s t h e sum  t h e TRSS  times  a  Jordan  al.  distribution  tensorial  by  et  t h e TRSS  of  t o make  calculated be  angle  three  Jordan  which  t o t h e MRSS  stress  MRSS  by  whereas  (TRSS)  t h e 4 5 ° cone  the  can  reported  or  from  t h e MRSS given  t h e MRSS t h e  the value  i n Figure  generation  9.4  [CRSS  reported i n which (yield)],  by M i l v i d s k i i  (MB)] and ( c ) t h e  and CRSS  131  Figure  9.3  Total resolved shear s t r e s s (TRSS) contours in MPa for the 45 cone a n g l e c r y s t a l . Compare the large s t r e s s l e v e l s of the TRSS w i t h t h e s t r e s s l e v e l s f o r same c r y s t a l s h o w n i n F i g u r e 9 . 1 ( c ) f o r t h e VMS and F i g u r e 9 . 2 ( c ) f o r t h e MRSS. /  4  C o n t o u r s o f t h e MRSS (MPa) i n e x c e s s o f : ( a ) CRSS (yield) ; ( b ) CRSS (MB) ; ( c ) CRSS ( M B T e ) . S h a d e d r e g i o n s i n d i c a t e a r e a s i n w h i c h t h e MRSS i s l e s s than t h e CRSS.  133  for  Te-doped  areas It  crystals  i n the figures  can  most  be  of  seen the  surface.  The  shoulder  a t A.  shown  in  crystal  where  to  Figure  the  largest  except  t h e MRSS  in  with  does  than the  occur  i s always  Doping  (MBTe) ] .  the effective  i s less  stresses  T h e MRSS  (b).  where  MRSS  crystal  resolved  in  not exceed  the y i e l d  a  greater  Te  stress  centre  in  The  and  near  the  below  the  t h e CRSS  (c) gives t h e CRSS  i s zero. stress in  region  than  shaded  (MB) a s  regions  (MBTe)  i n the  comparable  Appendix  in  Figure  of  7.1° ( F i g u r e  XIII.  9.4  with  Effect  The  effect  stresses  planes in  associated  the  crystal  the figures 9.7(a),  a ring  with  are  the  shown  in  similar  cone  angles  patterns  of the crystal  are given  to that  shown  a cone  angle  with  9.6(a-c)).  Angle  o f cone  on S t r e s s  angle  from  direction  higher  mm  from  cone  s t r e s s e s which  i n Figure  The v e r t i c a l  angle  low s t r e s s e s  on  the i n t e r f a c e ,  the horizontal  smallest with  distribution  i s shown  the i n t e r f a c e .  i s [ 0 1 0 ] and f o r the  Symmetry  on t h e MRSS  i n ( a - d ) a r e 7.5  i s complex  with  f o r other  show  the exception  i n ( e ) i s 8 mm  distribution  plots  t o the growth  The p l a n e s  Figure  and  distribution  These  o f Cone  perpendicular  plane  shear  of stress  in  9.1.1  directions  9.5.  Plots  by  that  specific  maximum  in  indicate  [CRSS  (a) .  The  e).  i s substracted  four-fold  and t h e  [100]. the  at the centre  exhibits  9.7(a-  direction  direction (7.1°),  planes  In  stress  followed symmetry.  134  I I  O-O-O-O-O-D <J> O A A A Q o o A A A LJ) 6 O A A • O o o AO o A 6 oo •o•oo o• o o o ' o \o o o, 6 o o oo o o o. o o o o o o o o I  I  I I I  I V  O 0  •  (III  M  (Til ( TT I  I  I 01  CI I0 3  T  03  ( M l  (T T  0]  (II.  COT  I  1  I  ( I l l  COM]  (III  tO I  (Til  (Oil]  ( i n  II  I]  x  V  ( TT (Til  o o o o o o o o o o ( M l I o o o\>. o o o o o o o \ o o o o o o o o o o o 6 o o o o o o o o o o o o. o o o o o o o o o o o o o o. <p o o o o o o o o o o o o o o. o o o o oo o o o o o o o • \ o o O o o • • • 1• c> o o o o • • o o o o o o o o o o o o • • • • o o o • • • • • o o o o • • • o A 1 • • • • • o o o o o o o o o 1 o o A • • • • • o o o o o o o o1 1 o A A o o A • • • • • • Ao o o o o o 1 1 o • • • • • • • o o o o o? o 0o o o 1 ?1 o A • • • • • • • •A ?1 A • • • • • • • • •A A • o o • o1 1 A A A A -Oo--o--o-o--o -ooo-•oo-o -0-0- 0-•0-0-  I  on  rT o n  [ToT ] l I  OT]  N  o-o1  I1  1  1  1  1  Figure  9.5  I  S l i p mode d i s t r i b u t i o n i n t h e ( 0 1 0 ) p l a n e cone angle crystal corresponding to d i s t r i b u t i o n shown in Figure 9.2(c).  f o r the 45° the MRSS  135  Midway ring  between  with edge  The  stress  at  of  the  at  in  For The  in  the  Figure  9.6(a)  central this  at  producing  across  the  stresses  at  stresses  at  about  eight  later  that are  stress  cone  angles  central the  the  centre mid  times the  fields  radius larger  qualitative  general. are  region  region,  This  the  are  is  in  total  as  towards at  9.7(b)  the  in  area.  the  ring  simpler.  stresses  surface  is  9.6(a).  is  Figure  low  at  irregular  Figure  and the  Outside  edge  edge  of  shown  of  are  Figure  stress  cone  angle  times  larger  than  the  the  minimum  characteristics because of  stresses  in  W-shaped  p o s i t i o n . At than  higher  largest  three  independent  value.  what  the  in  At  stress  with  a  the  about  centre.  a  For  A  with  stresses  characteristic  diameter. the  the  the  distribution  increases  the  maximum  to  at  shown  is  i n t e r f a c e . The  due  crystal,  the  a  than  the  is  symmetry  rapidly  at  from  cones  half  to  there  symmetry.  consistent  stress  45°  angles  than  the  the  includes  cone  mm  the  circular  stress  larger  and  is  wafer  circular  larger  stresses  of  the  increase  times  7.5  angles  30°  almost  larger  up  (e),  cone  both  times  build  half  the  For  For  upper  which  region  wafer. five  area  at  of  exhibits  pattern  higher  with  exhibit  five  with  higher  edge  stresses  is  complex  crystals  (c),  edge  the  which  the  This  the  and  stresses  wafer  center  contours  here  the  centre.  observed  centre  minimum  the  the  the  the  the  edge  values. of  9.7(d)  to and  distribution (65°) the  the  minimal  stresses It  symmetry  symmetry  thermal  the  starts  is  shown  presented  patterns  gradients,  are  of  the  providing  136  the  temperature  (radius,  boron  The is  to  shear  stress  CRSS,  the  cone and  MRSS  as  shown  excess  of  other  cone  (Figure W  are  is be  symmetry  in  the  distributions  shown In  the  cone  Figure  the  the  in  shown  the  the  CRSS of  the  the  in  the  For  angles  zero  are  as  symmetry  cone  is  resolved  taken  than  than  This  is  larger to  crystal  critical  angle.  results  the MRSS  central  Figure  obtained  so  9.9(a)  that the  centre  is  for  a  45°  this  figure  that  centre  than  the  The  direction.  stress In  the  no  be  7.1°  9.8  for  the  MRSS  has  eight-fold  from  the  and  45°  9.9(b))  by  cone  or  four-fold angle. It  in  stress  along  directions  the  the  central  symmetry  patterns  the  eight-fold  symmetry  as  this  case  the  observed  in  also  levels  in  the  In  can  exhibits  symmetry  increases  distribution [110]  distribution  dislocations angle  mm  9.9(a-b)  the  evident. has  2.5  (Figure  (Figure  replaced  will  edge.  and  cone  symmetry the  cross-sections  angles  (Yield)  As  9.9(c)  on  diameter  zero  evident. in  for  MRSS-CRSS  below  Figure  the  wafer.  the  cone  larger  greater  Similar  dislocation  on  the  always  For  is  the  in  (Yield)  similar  9.6(a).  angle.  across  The  region  in  cone  very  (Yield)  regions  9.9(c)).  centre.  is  on  conditions  unchanged.  above  CRSS  is  growth  dislocations  depends MRSS  the  angles.  shape  will  CRSS  stress  interface  the  of  the  and  remain  stress  When  stress  Figure  the  65°  The  a  in  etc.)  excess  symmetry  peripheral the  local  angle  the  linear  distribution  (MRSS-CRSS).  (Yield)  for  and  the  remains  thickness,  residual  smallest  and  oxide  density  related  in  profile  be  three the  times  radius stress  shown  larger depends  starts  to  137  igure  9.6  Contours i n MPa of t h e MRSS i n e x c e s s o f ( a ) CRSS (Yield) ; ( b ) CRSS (MB) a n d ( c ) CRSS ( M B T e ) . I n m o s t o f t h e c r y s t a l t h e MRSS i s l a r g e r than the c r i t i c a l values. T h e bump shaped contours (A) g i v e complex stress distributions i n perpendicular cross-sections.  138  a  139  140  C  141  d  gure  9.7  MRSS c o n t o u r s (MPa) in cross-sections perpendicular to the c r y s t a l a x i s f o r f i v e cone angles. (a) 7.1°, (b) 3 0 ° , (c) 45°, ( d ) 54. 7 ° , ( e ) 65 . S e c t i o n (a-d) a r e 7.5 mm from the interface and section (e) i s 8.0 mm from the i n t e r f a c e . The horizontal direction corresponds to the [100] d i r e c t i o n and t h e v e r t i c a l d i r e c t i o n c o r r e s p o n d s to the [010] direction.  142  Figure  9.8  C o n t o u r s o f t h e MRSS-CRSS ( Y i e l d ) (MPa) f o r t h e 6 5 ° cone angle crystal. MRSS contours a r e shown i n Figure 9.7(e). A t h i g h c o n e a n g l e s MRSS l e v e l s a r e larger t h a n CRSS a t t h e c e n t r e a n d o u t s i d e p a r t o f the wafer.  a  144  145  Figure  9.9  S t r e s s c o n t o u r s (MPa) in cross-section 2.5 mm from the interface. (a) and (b) in a 7.1 cone angle crystal, ( c ) i n a 45 cone a n g l e crystal, ( a ) MRSS c o n t o u r s , ( b ) a n d ( c ) MRSS-CRSS ( Y i e l d ) . F o r t h e 7 . 1 ° cone a n g l e t h e r e i s e i g h t - f o l d symmetry a t the c e n t r e i n (a) which i s not seen i n (b) b e c a u s e stress levels a r e l e s s t h a n CRSS. F o r t h e 4 5 ° c o n e a n g l e t h e r e i s four-fold symmetry.  146  drop  closer  addition result  to  the  cone  stress  larger  directions.  the  areas  Similar  than  in  any  remains  low  of  stresses  low  conclusions  even  other  close are  were  direction  to  the  present  obtained  ;  in  edge.  in  with  As  the  a  <110>  the  higher  angles.  In  a l l  the  characteristic symmetry  maxima  present.  This  This  i s more  local  are  associated with  9.1.2  Effect  In the  minima  Table  in  crystals  gradients  9.2  The  in  an  the  in  i n the s l i p  and  9.7(d),  and  <110>  where  the  symmetry  distribution  It i s also  i n t h e <100>  common  eight-fold  <100>  eight-fold  eight-fold 9.10.  one  wafer  the  Figure The  is  of the  observed  <110>  at  that  directions  mode.  Conditions  the e f f e c t  of  during  having  i n the boron  the thermal  growth,  a  45°  oxide  cone  conditions  stress angle  and argon  fields for  g a s , as  on are  different listed  in  .  calculated  tal  for a  d).  The  gradients  a change  distribution  determined thermal  apparent  t h e edge  to assess  of  minima  i n Figure  of Thermal  order  stress  at  with  with  there  edge  distributed.  i s shown  the  the  present  i s associated  mode.  considered At  are symmetrically  edge  slip  cases  i s always  directions.  the  center  range  of  isotherms with  temperature thermal  gradients  are observed  a small  distribution  to  curvature  be  i n the growing  i s shown very  crys-  i n Figure  9.11(a-  f o r low  thermal  flat  at the shoulder  of the c r y s t a l .  147  o ( 1 "" I )  • ( T i n  [  * (iT I )  11 o f ]  [1101 1]  0  (1 1 i )  LOT  •  ( T ""  1)  10 I I D  B  ( I7 I )  ft  ( I  1)  L 1 OT 3  e  (  C  *  /° ° * * ft  ^° °° °D •O ft  A  D  /  D  •  ft A  A  (  s a  • • • • O f t  o a a a a D f t A •  • • • • • • f t  a  S  B  fta  #  ft  A  o o o o o •  •  #  • • • • • • 4 * 8 9 9 9  e « d O A f t » B e  \ 9 0 0 0 9 0 0 f t f t  *  999Z99**  9  \a  0  0  0  9  0  0  *  •  a  B  *  *  * 9  •  •  A  ft  9  a  • • ° o • 9 a • ft ft ft • • #  Q  0 0  ft  9ft  a  a  a•  a"  a  a a a-  a  a  t I  T  03  \  a^  a  a\  • • • a a a a a a a 0 0 O 9 o o o a a a a a  e 0  • • • • • O O  ft  •  B  m  9.10  [ M O D  a a a a  a a  O 9 9 9 [ | 0 0 3  OOO  0 0 0 9 0  O O O  0 0 9 9 9 9 9 9 9 9  9 9 9 9 9  » 9 0 9 9  #  9^  9 9 9 9 9 ^  #  Figure  I 03  a B B a a a ^  • • • • 0 B ft ft • a • > • • A a ft ft 9 • • • ft ft • • • • 0 A ft • • » ft • • 0 ft A • 9 0 • • 9 9 9 9 0 9 9 * 9 9 9 ft ft ft ft ft * ft ft ft ft 9 ft ft 9 ft 9 • 9 ft ft ft 9 9 ft ft ft 9 9 ft ft ft 9 * •  <  o  ft  - ^ • • • • • • • f t f t B 9 9 9 9 0 0 0  (Til) ( TT I )  I I 3  O O  9 9 ft  9  [T  (III)  [ OT I 3  ft ft  ft  99  [ 1 0 13  0 I03  ft ft ft ft ft ft ft ft o ft o o o o s  ^ f t * ft ft A * A . * ft  CO  7 I I )  • ( I I I )  TOT ]  •  9  9  •  •  9  0•  .V  9  J  0  S l i p mode d i s t r i b u t i o n i n a ( 0 0 1 ) p l a n e corresponding t o t h e MRSS d i s t r i b u t i o n shown i n F i g u r e 9 . 7 ( c ) f o r the 45 cone angle crystal at 7.5 mm from the interface. The e i g h t - f o l d d i s t r i b u t i o n o f t h e mode a t the edge o f t h e s e c t i o n i s a s s o c i a t e d w i t h t h e e i g h t f o l d s y m m e t r y o f t h e MRSS.  148  a  b  149  Figure  9.11  Temperature distribution as a f u n c t i o n of external temperature gradient for a 45° cone a n g l e crystal. Thermal gradients in the boron oxide are : (a) 5 0 ° C / c m , (b) 1 0 0 ° C / c m , ( c ) 2 0 0 ° C / c m and (d) 4 0 0 ° C / c m . Temperatures are given i n 10 ° C . F o r l o w gradients isotherms are nearly f l a t and s l i g h t l y convex. For l a r g e r g r a d i e n t s i s o t h e r m s a r e c u r v e d and c o n c a v e .  150  Table  9.2  E f f e c t of Gradient  Ambient Temperature and S t r e s s  Temperature  Gradients,  on  Axial  Temperature  °C/cm |  Boron  oxide  |  Argon  50 100 200 400  For  a  temperature are  shoulder.  gradient in  the  Table  related. below  the  can  the  thermal a  and  stress  200°C/cm  MRSS,  the  is  the  also  AMRSS  of  50°C/cm  gradient  MRSS  in  in Figure the has  the  9.11(d)),  axial  The  (AMRSS),  listed.  relation  the  9.14(a)  the  listed  not  in  linearly  which  occurs  between  the  linear.  shown  different  The  gradients  are  are  near  temperature  axial  gradients  gradients  the  increasing  l a r g e s t MRSS  Figure  specifically  that  these  MRSS-CRSS a  curvature  imposed  in is  (Figure  gradient.  i s not  distribution  (b)  gradients  radial  that  table  the  shows  different  seen  the  gradient part  9.12(a)) higher  same  gradient  400°C/cm  be  MPa  0.59 2.55 5.12 11.91  400°C/cm  with  also  the  the  shoulder,  gradient  The  for  It  In  than  for  of  curved  figure  i s higher  9.2.  axial  The  crystal  25 48 84 98  gradient  clearly  AMRSS,  Axial  25 50 100 100  isotherms the  |  crystal  growing  in Figure  9.12(a)  9.13(a) and  (b)  (b). Part  (Yield). stress high  and  At  (a)  50°C/cm  below  and  and  distribution  stress  under  the  a  (b), for  gives  a  the  (Figure than  at  shoulder  151  Figure  9.12  S t r e s s c o n t o u r s (MPa) i n a C ^ p l a n e f o r a 45° cone a n g l e c r y s t a l grown with a 50 C/cm gradient i nthe b o r o n o x i d e , ( a ) MRSS c o n t o u r s d o e s n o t show t h e bump shaped s t r e s s d i s t r i b u t i o n below t h e s h o u l d e r i n ( a ) . The MRSS-CRSS (Yield) is positive only i n a few regions i n (b). 0 1 0  152  Figure  9.13  ( a ) MRSS c o n t o u r s and ( b ) MRSS-CRSS ( Y i e l d ) contours for a 200 C/cm g r a d i e n t i n the boron oxide. Units a r e i n MPa. MRSS s t r e s s a r e l a r g e b e l o w t h e s h o u l d e r i n ( a ) . T h e MRSS-CRSS i s g r e a t e r t h a n z e r o i n most of the c r y s t a l i n ( b ) .  153  Q  b  154  O-O-OO-O-O  6 oftftft6 9 oftftftcb o o ft ft a 6 6oftftD 6 6O DO o o o  o  III IV  o  o o o o o o o o o  6 o o o 6o o 6 o o o o o o o\> o o o o o o o o o o \ o o o o oo o o o o  V  s  s  o \ o o o o o o o o o o o o o o o  o o o o o o o o o o ^  6 o o  O O O O O O O O O O D\  O O O O O O f t f t f t f t f t f t f t f t D O O O O ft ft ft ft ft A ft ft A  ft  ft D N  O O f t f t f t f t f t f t f t O O O O O O D^J o o o o ft ft ft ft ft ft ft ft ft ft ft  6  •  ft ft ft o o o o o • • 6 o o • • • ft ft o o o o o o o • o 6 o• • • • ft ft o o o o o o o <!> o • • 6 o • ft ft o o o o o o o o 6 o• • • • • • • • ft ft o o o o o o 6 • • • • ft ft o o o o ft o o • • • o o o • • • • • D • Q ft ft ft O D O - f t . -ft- -ft- .ft-- f t - -ft. -ft--ft- • ft--ft. - o - - o - - o - - o - -o-6  ?° 1  Figure  9.14  ft • • • • • •  ( a ) MRSS c o n t o u r s . ( b ) MRSS-CRSS ( Y i e l d ) contours. (c) Slip mode distribution. Gradient i n the boron oxide is 400 C/cm. In (a) c o n t o u r s are similar to those obtained for a 200°C/cm gradient. Stress l e v e l s have d o u b l e d . In (b) o n l y a s m a l l a r e a i n t h e seed developed stresses less t h a n t h e CRSS. In ( c ) the mode d i s t r i b u t i o n i s similar to that shown i n F i g u r e 9.5 f o r a 100°c/cm g r a d i e n t .  155  disappears.  At  the  MRSS  the  thermal  the higher  configuration gradient  Figures  9.12  crystal  section  The the  slip  mode  9.14(c) modes  to  and come  Similar  operates  of  slip  From  the  relation  symmetry  o f t h e MRSS  and  Effect  Conditions  In  were  :  BG,  as  does  by  as  as  part  of  (b)  the  whole  i s t h e same  that  and  slip  this  is  gradient.  distributions In  shown  by  with  addition  the thermal  to  Figures  similar  temperature  components.  400°C/cm  increases  comparing  concluded  when  9.14(a)  zero.  configurations  not change to  seen  and  almost  below  for stress  stress  with  and  fourth  ;  the  gradient  Figures  is  9.7(c)  of  how  corresponds  sensitive  of  by  a  heat  factor  of  Four  different to  the c a l c u l a t i o n s are to  coefficients,  the  9.16  crystal.  to the three  50°C/cm  transfer  i n Figure  the  Coefficient  AG,  values  decreased  a r e shown  correspond  100°C/cm  of the heat  repeated  surface  Transfer  to determine  magnitude  results  the  among  in  at 400°C/cm  constant  expected  of the Heat  order  increased  the  the  shown  levels  i t c a n be  of  level  50°c/cm  (Yield)  stress  MRSS is  gradient  100°C/cm  this  100°C/cm  This  9.13(a)  9.15.  9.1.3  the  at  are  The  i s operative  value  modes  from  a  similar  similar  increased  For  which  from  independent  increases.  h a s MRSS-CRSS  9.5.  Figures  i s similar.  9.14.  mode  which  gradients,  the c a l c u l a t i o n s  transfer 1.3  respectively.  f o r the temperature profiles heat  the  The  profiles  are presented  transfer  ambient  coefficient  ;  coefficients  temperature.  at  three and The  156  CO 1 0 ]  Figure  9.15  MRSS c o n t o u r s i n MPa in a cross-section at 7.5 mm from the i n t e r f a c e f o r the c r y s t a l shown i n F i g u r e 9.14(a). The symmetry i s s i m i l a r t o t h a t shown i n F i g u r e 9 . 7 ( c ) f o r a 100°C/cm g r a d i e n t .  157  temperature shown,  profiles  are very  The  is  temperature At  the  similar  largest  coefficient  a t t h e s u r f a c e and c e n t r e l i n e ,  effect  junction  5°C. At  the seed  the  radius  at  t h e s u r f a c e and i n t h e bulk  temperature  The two  heat  (a)  corresponds  the  MRSS-CRSS  varies  a  not  transfer  The  change  change  i s  i s more  there  in  coefficient.  transfer  crystal  does  only  a r e shown  coefficients  t o t h e MRSS (Yield).  below  small,  pronounced  i s controlled  i s mixed  As a r e s u l t ,  substantially  20  values.  The  the crystal,  strong  value  o f MRSS  heat  transfer  distribution  effect  on  *  in  span  1.3  and h X  the  i t can the  9.17  and  control  a change i n modify  a n d 9.18 1.3  the  be  whole  for  values  seen span 53  that of %  the  respectively,  (b) corresponds  maximum  represents  of  the  heat  of the  to  MRSS stress  transfer original  temperature.  changing  the heat  the s t r e s s  field.  i n c r e a s e s from coefficient (Figure  h /  distribution  the shoulder,  about  i n Figure  Comparing  at the corresponding  Inside has  effect  material.  heat  transfer  and heat  In t h e b u l k  fields  transfer  coefficient values  not  field.  stress  occurring  the  i s small  coefficient  end.  i n the heat  seed  the seed  transfer  seed  in  the temperature  at  heat  surface.  change  the  t h e change  crystal/cone  because  the  at  are  i n F i g u r e 9.16.  of  observed  follows  approximately  to those  which  9.17(b)  transfer Note  that  0.05 MPa t o 0.16 MPa. does  not  and F i g u r e  change  coefficient t h e minimum  Increasing the  the  9.18(b)).  dislocation This  c a n be  158  0  Figure  9.16  I Position Relative  To  2 Interface  3 (cm)  T e m p e r a t u r e p r o f i l e s a t t h e s u r f a c e o f 45 cone a n g l e crystals for three (liferent conditions. Curve H corresponds to the temperatures calculated u s i n g the o r i g i n a l heat t r a n s f e r c o e f f i c i e n t v a l u e s . In c u r v e s H / 1.3 and H x 1.3 the original heat transfer c o e f f i c i e n t v a l u e s w e r e d i v i d e d a n d m u l t i p l i e d by 1.3 respectively.  159  Figure  9.17  Stress contours in MPa d e r i v e d from the temperature f i e l d obtained using heat t r a n s f e r c o e f f i c i e n t values 1.3 l a r g e r than o r i g i n a l values. ( a ) MRSS contours, ( b ) MRSS-CRSS ( Y i e l d ) .  160  Figure  d e r i v e d from t h e temperature 9.18 S t r e s s c o n t o u r s i n MPa values f i e l d obtained using heat t r a n s f e r c o e f f i c i e n t o r i g i n a l v a l u e s . ( a ) MRSS 1.3 t i m e s smaller than ( Y i e l d ) . V a r i a t i o n s o f only contours. ( b ) MRSS-CRSS maximum MRSS v a l u e s b e t w een 20 % a r e o b s e r v e d i n the (a) and F i g u r e 9.17(a).  161  accounted crystal which  for  stresses  to  and  of  dislocation  9.1.4  the  two  and  the  boron  crystal the  heat  temperature  velocity,  heat  section  transfer  stress profile  field. in  Increasing thermal  profile  or  shown does  i t was  the  In  boron  the  thermal the  considered  the  net  coefficient  on  the  the  30  an  gradient  and  the  argon  the with  growth  crystal.  variation  of are  on  together  the  in  affect  In the the  temperature  important  effect.  markedly  changes  calculations the  height  point  on and  pressure  each  changing  has  depend  the  the  *  the  diameter  substantially  that  thermal  and  may  profile,  inside  a  crystal  surrounding  and  and  entirely  the  dimensions,  not  media  oxide  at  coefficient,  that  those  almost  crystal  at  field  shown  Profile  media  mold,  determined  temperature  decreasing  i n both  the  transfer  surrounding  stresses.  in  crystals  coefficient  the  coefficient,  higher  coefficient  composition  is  i t was  However  the  from  depends  temperature  gas  heat  the  in  resulting  interrelated  into  profile,  determine  previous  are  the  the  This  the  Temperature  crystal  profile  transfer  surface.  are  transfer  transfer  input  layer,  are  crystal  i n the  i n the  when  transfer  These  the  gradients  negligible  heat  heat  i n heat  heat  variables  However  oxide  the  field the  two  like  pressure.  is  temperature  These  factors  known,  in  thermal  effects  Non-linearity  the  gas  the  of  in  both  increase  the  effects.  CRSS  When  temperature  crystal.  the  higher  in  increase  opposite  parameters,  surface  the  distribution  Effect  The  many  two  the  increase  from  temperatures.  result  on  the  resulting  leads  lower  by  the gas  was  temperature assumed  to  162  vary  linearly  may  not  be  measured  In  the  along  order  imposed  distance case,  9.3  as  the c r y s t a l  the s o l i d / l i q u i d  shown  by  growth  conditions  Gradients,  of  and  listed  the  Ambient  °C/cm  2  |  3  140  200,  z <  1.0  100,  z >  1.0  100 , z <  1.0  50,  z >  1.0  45 , z <  1.0  90 , z >  1.0  Argon  |  of the  stresses  i n Table  1  B 0  This  profiles  of n o n - l i n e a r i t y  Run I  interface.  temperature  temperatures  of N o n - l i n e a r i t y on S t r e s s  Temperature  the  the importance  profile,  f o r the four  Effect Profile  from  surface.  to determine  temperature  calculated  Table  with  AT 1  are  9.3  Temperature  in  B O3. 2 °C  100  350  100  350  50  200  180  Note  B o r o n o x i d e t h i c k n e s s = 25 mm ; a r g o n p r e s s u r e = 30 a t m . ; r a d i u s = 20 mm ; l e n g t h = 10 mm ; cone a n g l e = 3 0 ° . z is the d i s t a n c e from the s o l i d / l i q u i d interface.  163  The  calculations  taking boron  into  account  oxide  layer.  temperature in  3  gradients  The  MRSS  Comparing  shoulder stress  9.19  and  in  2  concentration shoulder  as i n c a s e s  is  1.6  about  (c)  occurs  Figure value  1  cm  patterns the  MRSS  direction  than  seen  are  f o r 4,  as  MRSS  The  than  large  and  9.19(a-d). stress  below  the crystal  the  i n (b)  3 and 4 i n F i g u r e (d)  rather  the  than  stress  below t h e  surface  and below  only  in 2  the equivalent  on t h e c o n e  occurs  is  i n 4 the  i n ( c ) f o r 3. T h e l o w e s t  stress  average  relative  as  In  i n t h e cone  the  difference  AMRSS  i n ( a ) . Comparing  surface  across  4  interface.  The  inside  3,  i s constant,  the  differ.  and  i n Figure  that  similar.  t h e cone  below  i n (d)  stress i n  t h e cone. the cone.  In The  i n ( c ) a n d ( d ) i s t h e same.  i n (001) planes  i s similar  t o t h e symmetry  encapsulant  stresses  is  2  similar  the melt,  are given  of the stresses  a r e shown  to  1 t o 3. T h e AMRSS  the lowest  symmetry  similar deep  i t  a t two p o s i t i o n s ,  the growth  and  cases  The minimum  larger  gives  to the encapsulant/gas  along  o f t h e minimum  The to  times  9.19(d)  close  1,  temperature  In 1 t h e g r a d i e n t  distributions  occurs  sets  the encapsulant.  cases  larger  MRSS  in  i n (b) i s twice  i n (a).  20 t i m e s  the  (b)  both  f o r case  for 1  about  close  two  of c o n d i t i o n s  across  f o r the four  (a)  distribution  pair  i s larger  i s larger  in  difference  profile.  the gradient  gradient  is  the  Each  the temperature  and  are divided  obtained  and a g r a d i e n t  i n Figure however,  f o r 1,  9.7(c) i s  2  and 3  for crystals of 100°C/cm. and 9 . 9 ( c ) .  four-fold  perpendicular of Table grown These  with  a  symmetry  The symmetry  at sections  9.3  of  5.0 a n d  164  165  Figure  9.19 MRSS ( M P a ) c o n t o u r s f o r a c r y s t a l g r o w i n g u n d e r four d i f f e r e n t c o n d i t i o n s g i v e n i n T a b l e 9.3. ( a ) Run #1 ; ( b ) Run #2 ; ( c ) Run #3 ; ( d ) Run #4. F r o m ( a ) t o ( b ) the AMRSS d o u b l e s . I n ( d ) t h e bump s h a p e b e l o w t h e s h o u l d e r does not appear.  166  7.5  mm  from  symmetry  crystal  is  stresses  to  a  cm  layer,  the  in  the  cone  layer.  This  thermal  discontinuity  The  result  stress  comparable  to  Comparable  9.21(c)  for  that  for  oxide  from  ambient can  this  be  (Yield)  1  does  not  observed in  Figure  *  the  occuring  and  2.5  with  shown  areas  small  are  obtained at  of  have  for  the  in the  areas 3  and  cone  has  the  areas  that  the  1  Figure  are  same  similar  cm  thick  9.13(b)  1  1  cm the the  larger  and 1  300  cm  *  thick the  encapsulant.  for for  case  2  1  thick  a  cm  is  obtained  in  cm  This  shows  the  boron  the  layer. depth  density.  stresses to  1  suppressing  9.21(b)  a  a to  in  in a  temperature  dislocation with  of  layer  across  crystal  also for  with  100°C/cm  gradients  the  oxide  comparable  noted  increasing  that  are  of  the  boron  generated  in  9.4(a)  the  a  is  in Figure  decrease  9.21(c)  to  with  length,  the  are  only  importance  of  Figure  cm  4  large  for  1  is  larger  distributions  3  to  stress  gradient  compared  shows  crystal to  for  It  distribution  layer.  zero  stresses  a  distribution  the  case  with  40  two-fold  cone.  9.4(a)).  when  2  results  (Yield)  present,  1  over  for  with  using  (Figure are  CRSS,  9.3)  obtained  encapsulant  stress  of  9.20.  stresses obtained  MRSS-CRSS  distribution  gradients  (Table  the  the  evidence  conditions  Similar  below  mm  in Figure  region  the  5.0  for  the  CRSS.  comparing  the  larger  the  position  thick  1  than  however,  When 2.5  less  below  4,  For  At  shown  (Yield)  9.21(a-d).  In  as  MRSS-CRSS  Figure  moved  interface.  i s observed  The  4.  the  those  less  of  profile In  Figure  in  the  addition  than  in Figure  the  i t  CRSS  9.21(a).  167  CO 1 0 ]  Figure  9.20  MRSS (MPa) c o n t o u r s i n a ( 0 0 1 ) p l a n e a t a d i s t a n c e o f 5.0 mm f r o m t h e interface in the c r y s t a l shown i n F i g u r e 9 . 1 9 ( d ) ( r u n #4). The f o u r - f o l d symmetry w i t h a s l i g h t t w o - f o l d symmetry i s o b s e r v e d .  168  169  c  21  d  MRSS-CRSS ( Y i e l d ) c o n t o u r s (MPa) f o r t h e four growth conditions shown i n Table 9.3 a n d MRSS contours shown in Figure 9.19 (a) Run #1 ; ( b ) Run #2 ; ( c ) Run #3 ; ( d ) Run #4 .  170  In  1,  however,  average stress  gradient of  is  the  the  in  3.  not  presented  the  shape  important  and  that  the  temperature  almost  shows  are  of  in  Is  This  which  results  effect  very  gradient  gradient  gradients  From that  the  twice  the  as  large  detrimental  as  the  effect  on  constant.  in  of  this  the  the  section  imposed  i t has  temperature  approximation  profile  in  the  been  of  boron  shown  profile  a  constant  oxide,  as  done  than  the  19 5 by  Dusseaux  stresses been  ,  obtained  shown  without  the  9 . 2  media,  Crystal  the  stress  model, real  As  changes  in  addition  oxide  the  contribute  R,  27.5  mm  ;  CL  :  a)  13.75  c)  55.0  e)  110  previous profile  level  temperature profile.  boron  In  thickness  thermal to  a  i t  alone,  profiles  substantial  has  in  the  decrease  Length  the  temperature  different  gradients.  the  not  are  levels.  Conditions  In  the does  which  variable  increasing  considering  stress  stresses  using  that  surrounding in  gives  result  temperature  conditions  and  inside  the  mm  mm  ; ;  b)  82.5  mm mm  ; ;  .  was  crystal.  the  27.5  d)  shown  important  i t i s very profiles  mm  i t  very  growing  gradient a  a  21.0  mm  section  is  in a  B,  variable  This  curvature important  which crystal  are  that  growth  for  of to  as  is  the  in  determining  both  the  use,  as  close  as  chamber.  ambient  the  axial  temperature input  in  possible  the to  Measurements  171  of  temperature  profiles  the  thermocouple  are  shown  The considered shows  arrangement  i n Figure  temperature  during  8.19  adjacent  MRSS  growth  shown  in  to the c r y s t a l  a r e shown  curve  reported  8.18. The  4  gives  using results  the  ambient  surface.  for  i n Figure  been  i n Figure  which  distribution  have  the  five  9.22(a-e).  crystal  Comparing  lengths  distributions  the f o l l o w i n g .  1)  In a l l cases  high  stress  the  part  of  outside  crystal, to  2)  high  The l o c a t i o n  surface  changes  with  For  are  crystal  length.  of  the  For crystal encapsulant  i s below  the  the  AMRSS  i s above  the encapsulant-gas  also  usually the  occur  narrow  at both  t o t h e ends part  the  ends  band  close  always  valleys  longer  of the c r y s t a l .  less of  crystals  surface.  of  the c r y s t a l the  crystal,  and o u t s i d e  stress  moves  the W  part of to the  distribution  exhibits a in  direction  the p o s i t i o n  o f t h e minimum The  at the  lengths  along  the centre  ends.  direction  For  at both  axial  The p o s i t i o n  surface  outside  shoulder.  way b e t w e e n  the radial  Close the  half  crystal.  crystal in  stresses  in a  present  i n the axial  AMRSS  and  shortest  i s always  the  T h e minimum  the  also  (AMRSS)  and i t s p o s i t i o n  the thickness  a t t h e c e n t r e and  interface.  o f t h e maximum  crystal  occur  the c r y s t a l . stresses  the s o l i d - l i q u i d  than  3)  (a)  levels  W shape  .  are closer to  173  4)  The v a l u e stress  o f t h e AMRSS  level  crystal  length,  slightly AMRSS  for (c),  ratio  =  mm  in  factor  are be  length  occurs  than  regions  close  similar observed  deformation  area  i n t h e cone  than  t h e CRSS  at the shortest as w e l l  (Yield).  i n the is  high.  AMRSS  the  restrained  (Yield)  length  changes  This  The  f o r(c) For the thermal  of the c r y s t a l . to  The  deform  quasi-plane  f o r the f i v e  9.23(a-e).  previously.  F o r t h e two  are  at  stresses.  i n Figure  as below  surrounding observed  of  the ends.  higher  sudden  a maximum  of the deformation. effect  the  i n the  a  of these  stresses  main  above  and g i v e s  result  the  to  considered  o f two  produces  this  i s more  gives  a r e shown  t o t h e MRSS that  a  the  o f t h e MRSS-CRSS  lengths  4,  at the midlength region  increasing  As  field  in this  contours  This  profile  curve  material  strain  (aspect  c o n t r i b u t i n g to the largest  crystal  the  o f 5 5 . 0 mm  are expected  coefficient  8.19  maximum  of the d i s c o n t i n u i t y  to the geometry  transition  of  transfer  and  The  i s the result  the c r y s t a l .  temperature  mm  because  with  for (c),  crystals.  stresses  of the  increases  value  length  1)  the i n t e r f a c e .  related  55  longer  =  higher  Figure  from  second  maximum  i n the temperature  In  the  is  for  surface  i n heat  media.  a  representative  crystal,  for a crystal  First,  curvature  30  reaching  surrounding  change  is  whole  diameter/length  encapsulant media  the  decreases  factors.  The  in  which  largest  results  In a d d i t i o n  f o r (a) there  t h e cone  The  crystal  where  is a  t h e MRSS  crystals  i tcan small  i s less  ( ( d ) and ( e ) )  175  the  top  CRSS  of  the  crystal  (Yield).  generation  When  in  undoped  values  are  very  9.22).  The  contours  are  similar  and  shown  to  in  The  with  be  to  modes  are  stress  a  III  and  It  the  distance  preferentially interface the  Effect  The two  growth  ways. and  symmetry with  length  the  crystal  level.  in  in  and  the  stress  MRSS  Te-doped  stress  III  is  cone  the  following  changes  are  on  length  different  reported.  maximum  modes  Length  crystal  greater  of  in  (Figure material  considered  at  in  length  The  not  most  area  the  regions  stresses  of  Mode  close  and  the  region  mm.  (a)  may  maximum  centre  27.5  be  frequent  of  central  than  to  (c)  at IV the  below  frequent.  Symmetry  stress  symmetry  described.  is  may  of The  alternatively  the  cross-sections are  function  9.24(a-e).  the  less  Stress  on  mode  and and  a  operating  crystal.  appears  surface  the  slip  slip  the  as  Figure  mode  The  in  region  crystal  symmetry  each  than  dislocation  the  in  MRSS in  slip  interface  of  are  the  Mode  other  First,  for  yield  the  shown  regions  The  the at  the  of  for  contours  (MBTe)  the  of  is  stress  appears  effect  the  obtained  when  mode  that  IV.  above  shoulder.  9.2.1  in  (b)  stresses  higher  9.23.  appears  from  stresses  considered,  encapsulant  also  (MB)  MRSS-CRSS  crystal  the  develop  those  the  specific  above  crystal.  to  indicates  associated related  is  slip  the  distribution  GaAs  obtained  Figure  in  CRSS  of  those  not  the  similar  operative  position  does  are  is  followed  Following  considered  presented  and  the  during  this  the  changes  176  •-D-O-D-O-O-B-B -B-B-B-B-B -fl-B-B-B-B -O-O-O-O-O-O-O-O-B-B-^-^-^-O-O / • • • • • • • • • • • • • • • • • • • • • • • • a a a o a o o o o o o o o ^ ^ o ^ i / . 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E CO ' IO CO 00 to as CO co CM  O e  O o o  ^—cu  (11-1  • >  6-000-0000-00-0-0-0-0-0-0-0-6  •  --1 E E  CO  I— I -  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  E E  0  J= 4->  — I  o  0  D  ^-^  in  0 0  0  u  1 — 4  -1-4  — •—  o o o n i i i D a a Q « ^ O o O O B B B B B Q D D n » /Oo°OOOOBBBBBBBD»  CO  >> 0  >  1  CO  1 4 — ->4  cCM  0  •-a-0-0-0 -O-0-4-4 - a - o Ja • • D 0 0 0 0 0 0 4  •-•  to E c cu i n  Si  ^ " Q O O O O O I I I I I I I I Q D D D D D D I O a 0 * / o ° o o o o o o o o i i i a o a a a a Q a o i /OOOOOOOOOOOOOOIIIODDODQQQI •-o o o o o o o o o o o o o o o o o o o m i o o o i 6-0 -0-0-0-00-0-0-0-0- 0-0-0-0-0-0-0- o-o-o-o- 0-0-0-0-0-0 0  £  >>•—•  • • • • • • • • o o o o o o o o o a a *  Q  •*->  CO 4-> CO  D Q G Q Q O Q D 0 0 0 0 0 0 0 0 0 4 0 4  /D  Q  / 4  CO  X  CO C7>  CU Si  3 to  •—)  CO  •rt  T3  CO  U  177  Considering fixed  the stress  i n the c r y s t a l  and  104.5  these  mm  from  symmetry  a r e examined  the cone.  cross-sections  are  The  during  at  symmetry  considered  in  formation.  In t h e model  the  given  and  i s assumed  time  MRSS-CRSS growth.  (Yield). It  is  considerations. hardening to  therefore The  dislocation  x  first  A  is  the  49.5  and  stress  be  the  interaction  varies  4  x  10  2 dislocation negligible  densities with  of  respect  growth.  effect  i s minimal  f o r normal  second  consideration  first step  and  i s related  during  assumed  that  growth.  between stress  two  i t may  to  steps.  reached  be  Because  stress  the The  at each  difference  and  final point.  4  i s no  x  i s the A  varies  involved 10  -2  in  MPa f o r  stresses  are  stresses  developed  that  hardening  a  the  consequence  hardening  i n a given  density  n  19 2  in dislocations  in  is  densities.  is partially  density  due  density, i . e .  These  concluded  dislocation  there  new  2  10 /cm  to the increase  the d i s l o c a t i o n  proportionally  to  and  the thermoelastic  during  The  Therefore  4  10  to  MPa  during  two  in  constant  -3  to the  The e f f e c t o f  of the d i s l o c a t i o n  between  in  a t any  i n the c r y s t a l  decrease  mm  dislocation  introduce  density. A c c o r d i n g t o Vakhrameev et a l . - 2 t o 4 x 10 MPa x m. T h e m a g n i t u d e o f s t r e s s  2  levels  changes  dislocation from  77.0  formed  level  stress  interaction  mm,  to  to hardening.  The  sections  proportional  to  the e f f e c t i v e  root  mm,  density  stress  i s related  interactions.  where  to  important  to the square  /~Ii,  the  four  relation  dislocation  However,  i s to decrease  proportional A  position  8.25  growth  is  given  after  effect  section  MRSS-CRSS  (Yield) by  of the  the  each i tis  increases values maximum  178  The all  specific  cases  the  1)  The  MRSS-CRSS  at  CL  (a)  13.75  (c)  55.0  :  stress  early  stage  encapsulant.  in  the  on  In  (b)  the  in  (a).  stress and  the  is  almost  symmetry  taken the  a  is  elongation  level  stress  has  the  of  in *  contained crystal  edge  of  low  and  is  has  (a).  with  In  four-fold  pattern  which  two-fold  minimum  symmetry  stress  area  and  the  which in  times  the  the  the  [110]  centre  symmetry  stress At  at  the  In  be  the  centre  symmetry  edge.  the  eight-fold  appears  can  than  where  The  the  Figure  higher  centre  (b).  area  centre  same  the  central  the  eight-fold  symmetry.  where  to  which  the  increased.  respect  is  position  centre  between  (c)  the  In  the  edge  an  crystal.  6  midway  by  the  the  to  section  finally  the  about  ring  the  and  at  on  in  the  between  In  is  depending  circular  a  in  corresponds  the  important In  shown  are  (a)  is obtained  edge  at  than 30  in  substantially  except  doubled.  slight  most  is  as  doubled.  definite  midway  the  not  decreased  has  of  at  are  stronger  a  density  In  ;  steps  crystal  the  follow.  ram.  four  changes  to  mm  82.5  shown  symmetry  centre  quite  the  more  ring  there  the  has  stress  the  eight-fold,  circular  periphery the  symmetry  Circular  27.5  (d)  at  sections  cone.  mm;  in  The  four  (b)  level  not  is  ;  stress  changes  edge  mm  the  The  stress  has  from  the  dislocation  The  considered.  when  from  the  is  distribution  four-fold,  moving  9.25(a)  The  the  growth  "1 MPa.  to  mm  in  of  section.  changes  8.25  distibutions  9.25(a-d).  than  (Yield)  Section  Figure  less  observations  again this  seen  direction  has in case  as  an with  179  a  180  181  182  gure  9.25  MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance o f 8.25 mm from the cone at four crystal lengths. ( a ) 1 3 . 7 5 mm ; ( b ) 27.5 mm ; ( c ) 55.0 mm ; ( d ) 82.5 mm. C r y s t a l r a d i u s 27.5 mm.  183  respect  to  the  decreased are  in  by  the  generated  dislocation the  is  largest be  with  stress  edge  will  as  symmetry  2)  CL  (a)  55.0  (b)  82.5  :  stress is  following  (a).  to  from  mm  in  and  the  (b)  the  and  at  the  symmetry  The  the  of  in  region  10  %  the  values  will  be  symmetry  a  this  it  dislocation  W  eight-fold  order  edge  maximum  of  final  determined  midway  W-shaped  branches  the  the  for  at  more the  narrow  ring  .  four-fold  The  with  cone.  cross-sections  9.26(a)  distribution can  and stress  minimum  stress  the  the  .  Figure  symmetry The  At  these a  growth be  and  show  dislocations  will  growth.  has  mm;  stress  symmetry.  edge,  The  mm  distributions  a  of  wafer  i t occurs  the  more  level  weak.  characteristics  circular  %  49.5  shown  four-fold  50  very  at  cone  In  also  The  since  end  From  will  stress  no  centre  centre.  of  the  the  this  (c).  wafer  noticed  Section  corresponds  by  be  in  the  variations  in  the  diameter.  as  At  during  At  the  a  hardly  will  The  the  (b).  that  high  be  reached  (d)  therefore  growth.  reached  along  dislocation  has  is  predicted  twice  the  in  In  and  distribution  stress  reached  than  wafer  further  density  distribution  direction.  whole  maximum  stress  may  [110]  be the  level  in  close  (b). to  rest  of  is  close  the  edge  the  stress  in  (b)  is  the  the to  central has  almost  mm  from  9.26(a)  interface.  The  central  surface  same the  49.5  Figure  the  distinguished.  level  occurs  and  at  at  the  has  The part  almost  centre  and  edge.  region  doubled circular  has  with  increased respect  to  throughout  the  184  185  Figure  9.26  MRSS-CRSS (Yield) distance of 49.5 lengths. ( a ) 55.0 27.5 mm.  (MPa) in a (001) plane at a mm from the cone a t two crystal mm ; ( b ) 82.5 mm. Crystal radius  186  section. be  The  final  determined  shaped same  as  dislocations  the close will  dislocation  to be  the cone, 10  %  Since  a  distribution  i n the outer  wafer  considerable  given  in  (1)  and  for a  %  W-  of this  with  section  f a r from  50  a  dislocation  i s compared  dislocations region  number  the  may  of the  symmetry,  When  section  case  of the W  four-fold  in this  the p r e d i c t e d  lower  the center  distribution.  predicted  in this  i n ( b ) . In t h i s  i n (a) with  i n the f i n a l  distribution  final  with  branches.  are generated  density  developed  i s predicted  the  be p r e s e n t  density  dislocation  by t h e s t r e s s e s  distribution height  will  maximum  t h e cone  higher  i n the  centre.  3)  Section  a t 7 7 . 0 mm  CL  ( a ) 82.5 (b)  The shown  stress  fold  in this  symmetry  stress  1 1 0 . 0 mm  9.27(a) case.  is  regions  t h e cone.  mm; .  distributions  i n Figure  observed  from  and  f o r a wafer (b). Similar  The d i f f e r e n c e s  expanded  midway  77 mm  and  between  in the  patterns  are that  (b)  there  centre  from  and  t h e cone a r e as  i n 2) a r e  i n (a) the f o u r are the  four  minimum  edge  i n the  <100> d i r e c t i o n s .  4)  The  Section  a t 1 0 4 . 5 mm  CL  110.0  :  only  corresponds  and  it  i s close  to to  the cone.  mm.  cross-section  cone  from  the the  available  longest  at  crystal  solid/liquid  this shown  distance  from  i n Figure  interface.  The  the  9.21(e) stress  187  a  188  Figure  9.27  MRSS-CRSS (Yield) (MPa) in a (001) plane at a distance o f 77.0 mm from the cone a t two crystal l e n g t h s , ( a ) 82.5 mm ; ( b ) 110.0 mm. Crystal radius 27.5 mm.  189  distribution shown is the  in this  i n Figure  reached  section  9.27(a).  I f growth  the d i s l o c a t i o n  features  i s very  similar  i s completed  distribution  i n Figure  9.27(a).  1)  an e x p a n d e d  four-fold  symmetry,  2)  a dislocation W  shape  close  to  branches in  this  for  If growth  along  considered,  1)  of  3)  sections,  The  at  in  average  at  of  This  direction  with  minimum  the  very  centre,  and  mm.  distributions  of  symmetry  a  density  of  the  during  stresses five  are  lengths  observed.  a t t h e edge is  of the s l i p  direction,  have  the dislocation  stress  c a n be  symmetry  end w i l l  density  3) a t 27.5  each  patterns  length  dislocation  the d i s t r i b u t i o n s  distribution  the stress  of the sections i s associated mode  with  an  o f t h e MRSS.  distribution  always  shape.  four-fold  interface  in  having  peak  half  the  crystal  present.  a W  be  following  In t h e r a d i a l shows  The  considered  The e i g h t - f o l d  i n the r a d i a l  wide  high.  will  the following  eight-fold  a  this  These a r e  the diameter  edge,  wafer  the  always  2)  the  equally  in specific  observed  across  a wafer  instead  distribution  distribution  when  at the t a i l  observed  a  to the  symmetry  between  5.5  mm  the top of the c r y s t a l  is and near  stronger 11.0  mm.  close It also  t h e cone.  to  the  appears  190  4)  A slight is  5)  tendency  always  Far  to a two-fold  accompanied  from  the  symmetry  by a f o u r - f o l d  ends,  at  symmetry  i s almost  symmetry  at the center  least  when  symmetry.  1 3 . 7 5 mm,  axisymmetric  present  with  the  slight  and e i g h t - f o l d  stress  four-fold  symmetry  at the  edge .  6)  For c r y s t a l far  from  in  lengths  larger  the interface  the  <100>  directions.  than  have  directions  The a r e a  5 5 . 0 mm,  t h e minimum  rather  where  cross-sections  this  stress  than  in  minimum  levels  the  occurs  <110>  i s very  smal1 .  9.3  Crystal  Conditions  Radius  :  R,  4 0 . 0 mm  AP,  effect  40  mm  on  is  determined  surrounding increased. Figure  In  ;  ( b ) 4 0 . 0 mm  (c)  ;  ( d ) 1 0 0 . 0 mm  8 0 . 0 mm  this  stresses  that  crystal case  the crystal  radius  ;  from  and d i s l o c a t i o n  the temperature has  not  changed  the  temperature  2 7 . 5 mm t o  distributions  profile when  .  i n t h e medium  the  profile  i s  radius  is  given  in  8.19 ( c u r v e 4 ) .  Values are  2 1 . 0 mm  ;  crystal  assuming the  B,  ( a ) 1 0 . 0 mm  of increasing  internal  CA, 3 0 ° ;  30 a t m ;  CL  The  ;  shown  f o r t h e MRSS  i n Figure  at the four  9.28(a-d).  crystal  T h e MRSS  lengths  distributions  considered follow  the  191  same  general  radius,  with  stress,  as  those  differences. the  long  explanation  usually  position there  i s an  plane  strain  given above  stress  increasing  the  radius  indicating  that  the  to  the stress  %  MRSS  increases  increases  The  in  due  The  rather  to  crystal.  maximum If  the  this  crystal  the  second  the  the  i s consistent  surface.  position  there.  when  of 1  section.  f o r the larger  45  ratio  encapsulant  prevailing  level  last  crystal  t h e maximum  occurs  difference  midlength  contribution  conditions  the higher  the  i s that  an a s p e c t  the  the  the smaller  process,  This  in  with first  growth  crystal.  with  extra  The  i n ( b ) ,with  occurs  coincides  observed  whole  i n the smaller  the  stress  is  two m a i n  i s 4 0 . 0 mm  2  with  as  considering  crystal than  patterns  quasi-  difference  In t h i s  stresses  case  by  40 %  with  the  levels  has  proportionally  radius.  The been  effect  explained  increasing  fields  radius  the r e s u l t s  the s o l u t i o n s  considered stresses. made  on  The  a  with  similar).  For  to  possible  could study  cause  c a n be  crystals  of  length.  same In  comparison  on  stress  considering field. of  result this  on  using  aspect this  pointed  of out  temperature  from  the  limitations  In  the  gradients  observed two  ratio,  with  lengths  present is  criteria. or  first  effect  both for  the  on  I t can  between  investigation  obtained  crystal  As  effect  on  effect.  the  the  radius  thermal  for  made  (The c o n c l u s i o n s  the  by  thermal  of radius  comparison  i s chosen.  This  employed  t h e same  the  radius  f o r the e f f e c t  the e f f e c t  f o r two  crystals method  as  the  literature  are contradictory.  investigation  be  increasing  i n the  the  previously,  of  of  two  second  methods a r e which  the  192  maximum the  stresses  length  40  mm.  40  mm  i s 55  radius  crystal  For  radial  the  ; axial  function  corresponds  (3)  i s below  (3)  in  thermal large  field  change  change  case  The  curve  constant cannot.  A  observed  bodies  the  than  in  gradient  radial  results  in  cylinders  two  incompatibilities when  radial  Figure  gradients Curve  (1)  (3) f o r t h e the  that  the  that  the  f o r by  the  radius  on  and  accounted  curve  (2) and  concluded  the  in a  of  crystal  i t is  stresses a  concluded  different with  considered  larger that  has  a  In  cylinders  ;  in  for free that  radius  the  cubes  i t  expansion. same  This  the e f f e c t  a  generates  creates  radius.  In t h e  different  cylinder  components  i t is  of  of the c r y s t a l .  geometries.  in  strain  effect  gradient  develop  gradient  present  seen  in  that  i n radius  the  geometry  cubic  can  From  clearly  mm  average  i s between  i t c a n be  for  to the s p e c i f i c  in  gradient  shown  figures  be  80  and  (2) and  (1)  cannot  explanation  incompatibilities the  curves  changes  f o r the  field.  stresses radial  by  axis  are  is  d i r e c t i o n s are  interface.  curve  observations  in stress  crystal  i n the  (a) and  length  i n two  the  and  observed  i s not i n f l u e n c e d  cylindrical on  crystal  (1) i n  these  is related  effect  mm  the length  (b) f o r t h e r a d i a l  from  crystal  specifically  gradients and  radius  another  gradients the  mm  crystal  considered,  gradients  be  radius  along  distance  alternative  of  mm  calculated  I t can  i n thermal  An stress  gradients  of  ( b ) . From  also  crystals,  t o t h e 27.5  crystal.  F o r t h e 27.5  are d i f f e r e n t ,  is  f o r the a x i a l  a  mm  lengths  three  gradient.  9.29(a)  are chosen.  F o r t h e 40.0  the  considered  40  mm.  Because  length.  as  occur  of  radial larger i s more thermal  194  Figure  9.29  (a) A x i a l along the crystal axis and (b) radial thermal gradients as a f u n c t i o n o f d i s t a n c e from the interface for two crystal radius. Curve (1) for a radius of 27.5 mm ; and curves (2) and (3) for a r a d i u s o f 40.0 mm. Crystal lengths are ( 1 ) 55.0 mm ; ( 2 ) 40.0 mm and ( 3 ) 80.0 mm.  195 strain  i s included  expansion  from  (contraction from  the  follow  a  as  the  initial  reference  i n this  centre.  an  increases  degree  expansion,  calculated  temperature.  case)  The  strain  of  which  This  with  as  free  a  free  expansion  increasing  distance  i n c o m p a t i b i l i t y i s expected in  turn  will  generate  to  larger  stresses .  The  effect  cylinders using  of  of a constant  different  the plane  temperature  strain  field  radii  -  r  can  gradient be  approximation.  G on t h e s t r e s s i n  quantitatively  The s t r e s s  evaluated  components f o r a  T = -Gr  G O  radial  aE (r - r )  3  1 - V  aE <2r  - r ) 0  1 - v  aE O  =  z  G  (r -  2/3r ) Q  l  It the  c a n be s e e n radius The  that  the stress  components i n c r e a s e s  linearly  contours  four  with  r . MRSS-CRSS  considered  - v  a r e shown  similar  t o those  having  stress  distribution  i n Figure  obtained levels  of  (Yield)  the  9.30(a-d).  with  below  for The  the smaller the  MRSS-CRSS  CRSS  (Yield)  the  results  crystal,  in for  (a) the  lengths shown a r e  with  and four  areas  ( d ) . The crystal  197  lengths MRSS to  show  contours.  those  9.3.1  in  relative As  following analysing crystal  by  those  between  Table  9.4  sections (1.8  x  tries  at  for  distance  from  larger  radius. the  same  for  a  to  the  are  similar  different  lengths  correspond  to  crystals in  the  analysis,  the  smaller  smaller  smaller  ways  namely  growth  crystal  for  the  From  crystal,  the  the the  and each  conclusions  comparing  radius  the  during  radius.  and  with  two  crystal in  cross-section the  the  the  two  radii.  crystal  was  the  obtained (2.8  40  mm  the  crystal  x  R)  symmetry  in for  crystal.  during  Figure  Similar  second  four  done  distribution  the  for  in  of  mm  in  (MBTe)  studied  was  correlation  symmetries shown  similar  are first  results  following  reached.  the  77  with  in  analysis  obtained  a  which  R)  stress  at  is  very  Symmetry  crystals  cross-section  obtained  There  are  MRSS-CRSS  considered  the  From  were  of  cross-sections  following  conclusions  cone  The  symmetry  those  which  Stress  in  positions  fixed  to  on  symmetry  the  analysis with  Radius  length.  similar  levels  9.30(a-d).  before, a  and  distribution  examined.  radius.  the  of  stress  been  same  The  Figure  Effect  The has  contours  growth.  9.27(a)  and  This  the  27.5  Both  (b)  of  2.8  times  but  at  a  relative  distances  from  the  the  radius  distance  obtained  cones  case  way.  have the  for  is of  following the  two  from  is  given  Consider  crystal  corresponding  cone  c o r r e l a t i o n s are  mm  sections  this  distance  correlation  following  the  In  with  and  very  a  mm  relative  times  other  radius.  the  symme-  reproduced 1.8  in  similar  stress to  72  the  in the  section  198  Table  9. 4  Radius  Table  [mm ]  |  Distance from S h o u l d e r [mm]  Relative Distance | from s h o u l d e r  |  Figure  8.25 12.0  0.3 0.3  9 . 25  27 . 5 40 . 0  77.0 72.0  2.8 1.8  9 . 26  27 . 5 40 . 0  104.5 112.0  3.8 2.8  9 . 27  this  of  the  of  the boundary  thermal  thermal  field  between  the  interface.  alone,  correlation there  distance  that  determining  with  passing  through  history  regardless  the  actual  of  interface  actual  history  thermal  of the c r y s t a l  the  effect  i s determined  by t h e  of  a  On  fields radius.  but  crystal cone,  with  the  the  lengths. however,  i s important  results  having  from  the contrary, f o r  the  the wafer would  correlation  distance  symmetries  from  This  over  strong  f o r same  distance  symmetry.  t h e same  be  i s not observed.  the  thermal  stress  also  correlation  from  the  the  is  effect  distance,  to predominate  should and  the  actual  I f t h e symmetry  there  symmetries  This  i s with  i s expected  conditions.  correlation  suggests  correlation  field  crystals  relative The  Correlations  27 . 5 40 . 0  Because  long  o f Symmetry  in  in  sections  t h e same  symmetry  199  9.4  Growth  Velocity  Conditions  R , 27.5  mm  CL,  Temperature Velocity  profile (a)  the with  determine  thermal  and  growth  fields  the  effect  stress  for  the  three  9.31(a-c).  Comparing  decreasing  the  of  velocity Increasing  significantly  increases  The lower  i n the  crystal.  results  indicate  the  thermal  independent  of  primarily (0.04  cm/s)  effect the  of  interface  to  which  steady  only ;  calculations,  as the and  velocity  that  at  cm/s  of  the  thermal  the  of  are  the  shown  on  this  and  (c)  )  temperature  cm/s  and  effectively  not  the  case,  diffusivity  used  velocity. fields the  that  temperature  0.001  is  Figure  noted  and  The is  of  transfer  the In at  specific  complex.  temperature  neglected.  heat  is  9.31(b)  state,  repeated  in  the  of  on  temperature  radial  movement is  velocity  were  i t  thermal  to  transient  independent  4,  cm/s ,  The  (a),  and  0.001  of  cm/s.  velocities  the  curve  0.001  growth  (Figure  approximation,  result  initial  0.01  steady  comparable  state a  axial  on  the  effect  the  value  velocity  in  and  the  Above  (b)  considered  little  the is  growth  quasi  changes  velocity.  due  and  at  mm  8.19,  calculations  9.31(b)  is  21  cm/s  has  system  cm/s,  the  0.0001  B,  in Figure  changes  velocities  distribution.  gradients  of  Figure  as  0.01  fields,  velocities  mm  0.0001  (c)  To  55.0  In  field  solidifying the  present  the  crystal  200  201  C  Figure  9.31  Temperature f i e l d s f o r three growth v e l o c i t i e s (a) 0.0001 cm/s, (b) 0.001 cm/s and (c) 0.01 cm/s. T e m p e r a t u r e g i v e n i n 10 ° C . L i t t l e c h a n g e i s o b s e r v e d from (a) to (b). From (b) to (c) gradients have i n c r e a s e d . R a d i u s 27.5 mm, l e n g t h 55 mm, encapsulant t h i c k n e s s 21 mm.  •-%f\L f'U  202  surface, state  there fields  temperature the  i s cooling  control  which  shown  the  values  cm/s  the temperature  Thermal  crystal  the  0.001  doubled area  fact law  uniform  t h e same a s  diffusivity  that of  the  that  any  external  cooling.  In  there  of the c r y s t a l  to the three  are  given  at  0.0001  and  cm/s,  this  i s mixed  with  a  4  the  to  in  %  Figure  cm/s  f o r the  highest  the  thermal  Biot  with  those  values  more  interface  of  than  with  at  2 % for  lowest.  values  fields  9.32(a-c).  ; approximately  and t h e lowest close  and  by t h e f a c t  are small  t h e MRSS  the  Comparing MRSS  have  doubled.  minimum  In  MRSS  increases.  Conditions  temperature on  velocity  distribution  the thermal geometry.  using  in  environment  The  effect  the temperature  of  as shown  i n Figure  the  the thermal  measurements  8.19, c u r v e  growing  surrounding  218 al.  steady  to one.  values  as t h e v e l o c i t y  modelled  with  Newton's  9.31(a-c)  disappears  the  velocities  corresponding  the  dependent  high  of  close  addition,  The  at  and t h e s u r f a c e  MRSS  of  with  approximately  9.5  low v e l o c i t i e s  include  cm/s, t h e d i f f e r e n c e s  highest  At  i s complicated  contours  Figure  Comparing  must  has v a l u e s  MRSS  in  0.001  values  following  i n the bulk  The  cases. ;  are expected  analysis  the s i t u a t i o n  number  expected  comparable  surface  limiting  boundary.  qualitative  case  are  fields  moving  For  0.01  a r e two  4.  crystal  i s  the crystal  and  environment  reported  by G r a n t  is et  204  C  Figure  9.32  MRSS-CRSS ( Y i e l d ) (MPa) f o r the three temperature fields shown i n Figure 9.31(a-c) corresponding to three growth velocities. ( a ) 0 . 0 0 0 1 cm/s, (b) 0.001 cm/s and ( c ) 0.01 cm/s. In (a) and (b) t h e stress fields are similar. In (c) the stress d i s t r i b u t i o n and s t r e s s v a l u e s change.  205  The oxide  profile  i s modelled  bottom power  thermal  of  the  input.  linearly  f o r the d i f f e r e n t  assuming  oxide  This  is  makes  that  the temperature  independent  of  the temperature  proportional  to  thicknesses  the  at  thickness,  boron  t h e t o p and  thickness  gradient  of  for  across as  constant the  oxide  demonstrated  14 6-148 experimentally It  i s also  chamber oxide but  by  new  the  only  profile  40.0  boron  input  the magnitude  t h e shape  of  or of  gradients  increasing the  local  the temperature  i n the  the  boron  temperature  profile.  i n the polynomials  are p r o p o r t i o n a l l y reduced  oxide  latter  being  thicknesses the  F o r each  considered.  This  fitting  by a g i v e n  the  factor for  The  considered  maximum  thickness,  crystal  a r e 2 1 , 40  thickness several  radii  currently  temperature  examined  a n d 50  are  mm  used  gradients  27.5  mm  and  mm.  9.5.1  Radius  9.5.1.1  Boron  27.5  The  mm  oxide  Average  9.33.  power  the axial  conditions.  experimentally. are  the  reducing  the c o e f f i c i e n t s  growth  The  that  changes  not change  that  original the  increasing  thickness  does  means  assumed  gradients  temperature  Five  thickness  profiles  21.0  mm  ( a ) 66°c/cm,  (b) 33°C/cm,  (c)  (d)  profiles a r e shown.  17°C/cm,  considered The p r o f i l e  are  8°C/cm  given  labelled  in  Figure  G with the  206  Figure  9.33  Temperature p r o f i l e s a l o n g the c r y s t a l s u r f a c e i n the environment surrounding the crystal employed in the calculations. The profiles are derived from Figure 8.19, curve 4, r e p r o d u c e d as c u r v e G. The values of gradient are averaged in the crystal length considered. Encapsulant t h i c k n e s s 21 mm.  207  steepest which  gradient,  was  results  profiles  are  patterns  are  however, result  are  to  in  lowest  the  than  zero  stresses  crystal  9.5.1.2  the  in  the  were be  in  from  shown  the  (d)  Figure  8.19,  CRSS  crystal  with  and of  (Yield). located  (d).  curve  4,  the  The where  surface  area  the  above  highest the  axis.  Boron  oxide  thickness  AVG  (a)  58  cm/s,  (c)  19  cm/s  B  50.0  AVG  (a)  50°c/cm.  (c)  12.5°C/cm.  40.0  mm  (b)  29  cm/s,  (b)  25°C/cm,  mm  zero  with  interface  remaining  levels,  gradient.  stresses.  has  distribution  with  regions  for  Comparing  stress  the  crystal two  The  average  the  contours  stress  the  lower  temperature  9.33(c).  the  The  above  four  stress  Figure  cases.  with  to  for  The  that  four  regions  most  in  levels  (a)  shoulder  are  in  (Yield)  noted  stress  the  in  shown  9.34(a-d).  considerably  from  seed,  less  ;  G  can  lower  gradient  appear  i t  increases grow  as  MRSS-CRSS  Figure  similar  the  gradually start  in  decreases of  the  profile  9.34(a-d)  same  experimentally.  for  shown  temperature  Figure  the  determined  The  the  is  zero  These  regions  stress stress  At  the  which  are  with  non  stresses  encapsulant  a  regions  edge.  stresses  As  and  usually in  the  209  c Figure  9.34  d  MRSS-CRSS ( Y i e l d ) (MPa) for four temperature profiles with average gradients ( a ) 33 C/cm, ( b ) 17 C/cm, ( c ) 11 C/cm and ( d ) 8 C/cm. Radius 27.5 mm, length 55 mm. E n c a p s u l a n t t h i c k n e s s 21 mm.  210  The are  temperature  shown  in  Figures  encapsulant  and  temperature  profiles  each for  profile the  The  40  mm  (a)  profiles  21  only  mm,  the  extended the  encapsulant  For  in  a  be  low  and  stress  three  gradient  original  mm  for  profile  Figures for  for  the  9.36(a-c)  the  three  50  mm  two  for  the  thickness.  different  thermal  (b).  be  mm  compared and  that  the  9.37(a)  : 9.23(c) for  the  to  to  the  the  50  boron  distribution.  close  concentration  encapsulant  are  increasing  stress  region  the  40  case  average  increasing  to  40  the  each  (Yield)  the  of  relative  stress  thick the  radial and  case  the noted  the  in  to  figures  for  In  the  9.37(a-c)  effect  for  The  position  and  uniform  mm  oxide changes  i n t e r f a c e and  a  for  above  the the  surface.  in  valleys the  can  mm.  thicknesses  shown.  shown  9.35(a)  three  changes  of  region  the  the  It  thickness  shift  Figure  (a)  comparison  corresponds  in  In  MRSS-CRSS  Figure  9.36(a)  thickness.  are  (c)  observe  thickness the  are  to  50  is also  thicknesses and  the  For  the  shown  To  given.  encapsulant  (b).  considered,  for  thickness  Parts  are  both  and  for  encapsulant  results  encapsulant  (b)  also  mm  for  9.35(a)  in  is  21.0  profiles  encapsu1 ants, low  part  direction sharp  of  the  50  region  encapsulant  only  mm  in  mm  large  crystal  maintains  the  centre  thickness  below 5  the  which  peaks  concentration is  of  the  the  shorter  in  has  a  stress  the  W  shape  edge  of  and  Figure  shoulder than  the  is  with  the  small  stress  distribution  9.37(a)  crystal  low  extended  crystal. the  stress  because  length.  In  the  211  Figure  9.35  Temperature p r o f i l e s used i n the c a l c u l a t i o n s f o r two boron oxide thicknesses. ( a ) 40 mm, ( b ) 50 mm. The profiles are derived from curve G. The values of gradients given are averaged along the crystal length.  211  Figure  9.35  Temperature p r o f i l e s used i n the c a l c u l a t i o n s f o r two boron oxide thicknesses. ( a ) 40 mm, ( b ) 50 mm. The profiles are derived from curve G. The values of gradients given are averaged along the crystal length.  212  213  Figure  9.36  MRSS-CRSS ( Y i e l d ) (MPa) for three average g r a d i e n t s . (a) 58°C/cm, (b) 29°C/cm, (c) 19°C/cm. Radius 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 40 mm.  214  215  Figure  9.37  MRSS-CRSS ( Y i e l d ) (MPa) for three average gradients. (a) 50°C/cm, ( b ) 25 C/cm, (c) 17°C/cm. Radius 27.5 mm, l e n g t h 55 mm, b o r o n o x i d e t h i c k n e s s 50 mm.  216  The  absolute  encapsulant maximum  thickness.  both  zero to  thermal  the  region  similar  thickness, achieved  a  times mm  gradients gradient  GaAs,  with  when  increase  reduces  shows  i n the  by h a l f t h e  level,  the  t h e 21  mm  environments Without  resulting the c r y s t a l  is  detrimental.  large proper i n the and  of  large  thickness  the areas  because  of  of  that for a  encapsulant should  view,  less  be than  a  thicker  reasons.  To  get the  should  be  reduced  of  three the  In  of c r y s t a l  encapsulation  leads  of  only  thickness.  formation  thickness  stresses  gradient and  encapsulant  10°C/cm  point  zero  I t i s noted  mm  with  following  thermal  is  about  close  largest  f o r an  a 21  with  regions  The  independent  areas  practical  This  increasing  9.37(c)  with  considered  i n the c r y s t a l  decreases.  that  f o r the the  are  o f 12.5°c/cm.  gradient  From  gradients  t h e low s t r e s s  obtained  to obtain  damages also  from  i n Figure  This  thickness.  temperatures. of  an  the region  gradient  were  i s preferred  10°C/cm  50  the  critical  (Yield).  desired  expands  areas  i n order  CRSS  lower  and a g r a d i e n t  8°C/cm.  encapsulant  the  as  o f 50 mm  of  with  i s obtained  surface  gradient  eight  with  the thickness  thicknesses,  (Yield)  interface  thickness  the  changes  Doubling  profiles  encapsulant  MRSS-CRSS  stress  also  stress.  When for  stress  times  reduction  addition  are exposed  there liquid  with  in to  of low high  i s decomposition Ga  drops  which  to non-stoichiometric  melts  which  217  9.5.2  Radius  9.5.2.1  40.0  Boron AVG  The layer  oxide :  a)  results  (a)  thickness  21  14°C/cm,  (b)  for  thickness  Part  mm  of  the  21  (b)  to  the  profile  the  average  to  G/8  in  considered.  Figure  9.34(c)  and  CRSS  (Yield)  case  of  top  of  due  to  the  the the  in  mm  :  the  mm  the  thickness  40  not  mm  crystal,  the  two  (b)  26°C/cm,  (c)  15°C/cm,  (d)  9°C/cm.  9°C/cm  is  in  original  9.36(a-d)  the  not  contours Note  the for  that  shown  in  encapsulant  temperature  the  27.5  that  of  the  (b)  can  be  noted  zero  MRSS-  as  with  in  the  however,  the  stress.  This  is  crystals.  for  the  the  profile  Figure and  profile  results  and  mm  46°C/cm,  (Yield)  (b).  and  large  zero  and  because  it  as  oxide  noted  relative  shows  between  is  9.38(a)  the  40  boron  (14°C/cm)  It  crystal,  is  9.38(b)  G/4  lower  (a)  Comparing  Figure  are  a  9.38(a)  9.33.  Figure  profile  length  9.39(a-d).  gradient  Figure  crystal,  In  Figure in  labelled  27.5  same  with  Figure  profiles  mm  crystal. in  in  the  40  the  oxide  Figure of  average  in  27.5  MRSS-CRSS  gradient  9.33.  for  Boron  The  than  area  profile  Comparing  for  the  difference  AVG  shown  of  crystal  the  9.5.2.2  (d)  case  in  shown  these  crystal  in  are the  longer  that  (Yield)  (7°C/cm)  gradients  7°C/cm  MRSS-CRSS  mm  corresponds  mm  of  gas  gradients with  9.35(a). is  five  labelled  Figures mm  four  average  The  9°C/cm  times  smaller  G  9.39(a-d)  crystal  an  are  i t can  in with be  Figure those  observed  218  Figure  9.38  average gradients. MRSS-CRSS ( Y i e l d ) (MPa) for two l e n g t h 80 mm, (a) 14°C/cm, ( b ) 7 C/cm. R a d i u s 40 mm, b o r o n o x i d e t h i c k n e s s 21 mm.  220  c  Figure  9.39  d  MRSS-CRSS ( Y i e l d ) (MPa) f o r four average gradients. (a) 46°C/cm, (b) 26°C/cm, (c) 15°C/cm, (d) 9°C/cm. Radius 40 mm, length 80 mm, boron oxide thickness 40 mm .  221  that at  the concentration the midlength  27.5  mm  crystal.  In  addition  areas  with  instead  stresses of  the  i t i s observed  zero  environment  of  MRSS-CRSS  surrounding  i n the l a r g e r top  of  that  40  crystal  to obtain  (Yield),  the  the  mm  crystal  the  occurs  as  similar  i n the  relative  gradients  in  should  further  crystal  be  the  reduced.  9.5.2.3  Boron  Oxide  AVG  :  thickness  (a) 43°C/cm, (c)  In  order  maximum  to  value  comparable 52  mm  (Yield)  stress  80  mm  crystal crystal. crystal  this  encapsulant  i s higher From radius  than  these  length  a  40  mm  9.40.  The  case  the  surface.  the l a r g e s t  stress  The  with  length  a  length  is  f o r MRSSthat  crystal  the  9.41(a)  length  of  concentration  occurs  value  longer  value  obtained  to generalize  i n LEC g r o w t h .  stress  In F i g u r e  for a  a  period,  It i s observed  stress  and those  reached  The r e s u l t s  stress  The  has  crystal  the shoulder.  stress  above  crystal  i s 50°C/cm.  same  i t i s possible  level  growth  thickness.  below  results  occurs  stress  whole  i n Figure  occurs  In  the s i t u a t i o n s  crystal  the  for  f o r the  shown.  the  i f the  gradient  a r e shown  concentration  is  (b) 14°C/cm,  the encapsulant  distribution  above  of  to  made  and t h e average  CRSS  the  determine  are  mm,  7°C/cm.  considering  calculations  50  i n the i n the  with  t h e 27.5  the f i n d i n g  largest  the encapsulant  stress  shorter  to  at a  surface.  mm  most given  If this  222  Figure  9.40 MRSS-CRSS ( Y i e l d ) (MPa) f o r a c r y s t a l l e n g t h o f 52 comparable to the boron oxide thickness o f 50 R a d i u s 40 mm. Average gradient 50°C/cm.  mm mm.  223  224  Figure  9.41  MRSS-CRSS ( Y i e l d ) (MPa) f o r three average gradients, (a) 43°C/cm, (b) 1 4 ° C / c m , ( c ) 7 ° C / c m . R a d i u s 40 mm, l e n g t h 80 mm, b o r o n o x i d e t h i c k n e s s 50 mm.  225  position  i s half-way  largest  i n the whole  The are  along  stress  shown  in  encapsulant,  the c r y s t a l  growth  the  9.41(a-c).  gradient  relative  areas  27.5  crystals  distributions.  The  evaluation  summarized CRSS  i n Table  (Yield)  gradient values  of  the  the  radius  also  the  apply,  that  to the  40  radius  listed  This  as  relative as  more  gradient when  that  from  to  an  growth  each in  values  conditions is  also  increase  In  case  on  stress  For  It  an  of  are  increase  decreases  increasing  The  tabulated  to  the  o f MRSS-  o f 2.  level the  obtain  thickness,  gradient. due  mm  with  value  ratio  40  section  oxide  the  lower  used  the e f f e c t  to  radius.  due  change  stress  reduced  boron  favourable  and  the  in this  aspect  clear  considered  comparing  crystal  an  are present).  %  i s the  of  the l a r g e s t of  and  stress  the  143.7  shows  i s not  is  case  when  increment  encapsulant  the lowest  increases mm.  A  the  and  It  the  presented  function  stress  gradients  further  table  profile  decreases  conditions  encapsulant  a  in  stress  results  as  reported.  increases  favourable  zero  In t h i s  three  be  to c r y s t a l s with  i . e . thicker  radius  stress  9.4.  relative  stress  observed  the  thermal  correspond  is  with  i s given  condition  that  of  As  should  similar mm  the s t r e s s  period.  d i s t r i b u t i o n s f o r the Figure  length,  (or  the  in more  thickest  i n the c a l c u l a t i o n s , the the radius stresses  from  27.5  mm  with  increasing  a l l  conditions  linear.  comparison i n the t a b l e  of  the  shows  stress that  contours  similar  for  stress  d i s t r i b u t i o n s are  Table  9.4  Effect  Boron Oxide Thickness | [mm]  of Thermal  Conditions  AMRSS G  | R  - CRSS  Stresses  (Yield)  MPa 1  = 2.75  G G/2 G/4 G/6 G/8  7.02 3.76 1.78 1.03 0.65  40 . 0  G G/2 G/3 G/5  4 . 39 1 . 97 1 . 22  50 . 0  G G/2 G/3 G/5  3 . 26 1.14 0 . 64  21.0  on  cm  | R  = 4.0  cm  % change  9.65  37.46  2.76  55.0  1.02  56.0  6 . 77  54 . 2  3.21 2.01 1 . 02  62 . 9 64 . 7  5 . 61  72.08  1 . 56 0 . 47  143.7  227  obtained areas  for  with  following  9.6  similar  zero  values  stress  are  B = 21  mm,  G/6,  r = 2.75  2)  B  = 21  mm,  G/8,  r = 40  3)  B = 40  mm,  G/5,  r = 40  Pressure  :  and  AP, B,  2 atm 21  AVG  The through  effect  mm  ;  coefficient  changes  gradients  c a n be obtained  9.42(b-c)  observed  mm  crystals  similar with  the  ;  ;  mm.  27.5  that  the s t r e s s  atmospheres  i s similar  CL,  55  mm  ;  i n Figure  9.33  ;  (d)  pressure on  is  by  crystal  The  considered  compared  and  the  gas.  of  (Yield) effect  of  30  part In  are  model  of the  the  heat  equations  heat  shown  of pressure  transfer  in on  values  atmospheres. Figures  Figures the  four  with  the  For  9.34(a-c).  c o n f i g u r a t i o n f o r the four to the stress  the  pressure.  the stress  with  in  convective  root  comparing  pressure be  ll°C/cm.  the convective  the square  analysed  should  as  17°C/cm,  gradients.  a  ;  (c)  evaluation,  with  for  mm  (b) 33°C/cm,  f o r t h e MRSS-CRSS  f o r four  mm  profiles  between  numerical  results  R,  of pressure  9.42(a-d)  results  the  (a) 66°C/cm,  gas  coefficient  f o r the  The  in  instance,  ;  :  of  the e f f e c t  transfer  For  Composition  Temperature  2  given  1)  Gas  Figure  stress.  conditions  Conditions  used  of  this It i s  gradients  c o n f i g u r a t i o n f o r the  at  same  229  c .42  d  MRSS-CRSS ( Y i e l d ) ( M P a ) f o r f o u r t e m p e r a t u r e profiles with average g r a d i e n t s . ( a ) 33°c/cm, (b) 17°C/cm, ( c ) 11 C/cm a n d ( d ) 8 C/cm. R a d i u s 2 7 . 5 mm, length 55 mm. Encapsulant thickness 21 mm. Argon pressure 2 atm .  230  imposed the  largest  from in  30  is  not  results  pressure effect  the  (Yield)  presented  important  observed  as  in  component  of  atmospheres  reduces by  74  %  less  10  %  of  than  coefficient.  In  coefficient than  the  pressure  A may  *  i n heat  stress  this  level  stronger  effect affect  surrounding  the  assumed  changing  that  maximum  the  shown  does  This  calculations  of the  crystal.  pressure  In  the  the pressure  pressure  a  way.  affecting  from  as t h e  30  heat  to  2  transfer  convective  part  of  the  transfer  in  heat in  the  pressure  above,  explains  i s not  why  is  total  will  that  a  be 30  %  affect  the e f f e c t  of  significant.  i s expected  does  as  of  stronger  not s u b s t a n t i a l l y  profile  present  about  coefficient  9.1  temperature  area  effect  when  variation  in Section  is  the  reduction  coefficient  of  weak  convective  value  i n the c r y s t a l .  i n the present  markedly  with  I t was transfer  the  The  pressure  pressures  %  value.  i s included  the  of  total  case  associated  30  Reducing  the  increased  the e f f e c t  transfer  3  the  f o r i n the f o l l o w i n g  of pressure  low  considered  calculations  value  only  At  the decrease  stress  expected.  the heat  . At  decreases  9.42(d)  that  accounted  of  the  be  atmospheres.  gradients  Figure  present  pressure.  coefficient  change  the  the e f f e c t  root  stress  gradient  show  might  t h e model  square  less  here  c a n be  convective  in  o f 30  i n the largest  i s expected,  In  the  o f 20 % . F o r t h e l o w e s t  MRSS-CRSS  as  pressure  For the lower  f o l l o w i n g the decrease  The  at a  considered,  to 2 atmospheres.  zero  %  profiles  gradient  the order  with 25  temperature  because of  the  calculations,  not a l t e r  the  pressure media i t  was  temperature  231  of  the  environment.  the  pressure  the  crystal-gas  transfer  oxide-gas  by  The gas  a  is  the  be  the  but  coefficient  at  efficiency  to  heat  will the  be  boron  decrease the  at  the  gradient  gas.  at  the  described,  with  boron  oxide-  assuming  surface.  associated  at  increasing  coefficient  this  in  efficiency  be  i n the  qualitatively applies  will  When  coefficient  coefficient  surface  transfer  the  higher  effect  probably  situation.  transfer  also  This  oxide  actual  heat  transfer  net  and  heat  mechanism  transfer  heat  be  the  altered.  boron  in  can  only  The  encapsulant  not  changes  larger  change  convection  written  gas  of  surface  heat  not  interface.  the  may  interface  the  temperature across  changes,  in  manifested  This  In  this  a  this  free  case  process  the  can  be  as I/O  h  =  where  Nu,  Nu  Gr  k/1  and  =  Pr  are  respectively.  "k"  characteristic  length  heat  gas,  number  the  will  pressure  by  83  At  the  %  similar  when  of  If  the  the  '  thermal  the  encapsulant-gas the  In  heated  that is  the  heat  the  with  chamber  convective  gas  case  transfer  this  Prandl  heat  an  30  from  in  i s more  transfer  which an  Grashof  heat  2  the  like  the  exponent  to  is  behaves  coefficient  from  change  the  numbers  "1"  surface  introduced  this  pressure  surface  and  conductivity.  on  the  (9.1)  Grashof  horizontal  dependence  depressurizing  (k/1)  Nusselt,  density.  depend  of  Pr)  i t i s assumed  dependence  changes  (Gr  the  pressure  through  coefficient this  is  is extracted.  ideal  0.14  transfer 2/3. is  For  reduced  atmospheres.  important coefficient  than at  232  the  crystal-gas  main  heat  transfer  transparent across  to  the  changes.  transfer  case  heat  increasing  the  stronger  will  changes  The  the  i n the  the  with  the  gas.  effect  encapsulant  the  net  the  heat  pressure  the  of  of  pressure  oxide  is  pressure  increases  the  In  this  reducing  increasing  the  the  surface  could  in  result  and in  crystal  a  than  coefficient.  et on  above.  by  crystal  encapsulant  Emori  the  gradient  improved  effect  at  is  surface.  is  transfer  discussed  thermal  affected  distribution  of  boron  the  crystal  The  results  convection  because  directly  coefficient  convective  the  is  case  crystal-gas  the  stress  experimental  agreement  the  gradients across in  first  increasing  from  in  more  at  transfer  effect  the This  be  hand,  extraction  heat  across  other  gradients  in  Consequently  coefficient  convective  any  radiation.  the  heat  since  mechanism.  encapsulant  On  thermal  surface  al .  the  For  14  3-144  are  thermal  argon  in  gradient  gas,  reducing  2 the  pressure  the  encapsulant  thickness larger  of  and  vertical  the  wall  =  were gas  in  is  heat  was  (k/1)  reduces  not  pressure They  transfer For  have  In  1  /  thermal  this  to  neon  neon a  and  helium the  stronger  explained  case  gradient  140°C/cm  having  has  calculation  is written P r )  neon  For  coefficients  the  assumed.  (Gr  with in  the  160°C/cm  given.  measured,  helium.  equation 0.53  kg/cm  approximately  pressures.  coefficient  5  encapsulant  than  calculating  to  from  Changing  gradients  h  20  gradients  gradient.  gases  from  the  for  heat the  ;  the  gases largest  effect  results  the  on by  different  transfer heat  in  from  a  transfer  as 4  (9.2)  2 33  This The  equation  involves  differences  exponents  acts  very  good  the  different  present.  larger  heat  temperature  the  correct  and  pressure  affects  the  through  changes  encapsulant-gas coefficient  The  gradient  the  through  temperature the A  for  the  heat  as  transfer  show  a  crystal using  A)  mechanism.  an  the  above.  gives  The  gas  encapsulant  coefficient  from  at  heat  the  transfer  horizontal  wall  14 3 (Equation  (9.1))  gives  results  the  deviation is  from  and  possible  the to  shown  in  linearity  c o n s i s t e n t with From  measured  the  model  conclude  gradients  Figure  is  observed.  physical results that  9.43,  the  process and gas  the  from curve The  Emori B.  a  reduce  (curve  between  for  Figure  and  the  et In  a  basically  explained  across  the  transfer  is  9.43  different  transfer  of  in  to  the  oxide  without  tend  correlation  heat  the  crystal  will  gradient  heat  boron  the  of  length  the  made  in Figure a  the  reproduced  encapsulant  correlation  meaning  however,  were  (9.1).  calculations  in  is  Equation  coefficients,  is  with  introducing  surface.  values  wall.  coefficient  but  in  "1"  measurements  in  the  case  correlation  across  tendency  and  correlation,  transfer  the  the  as  numerical  pressures  Moreover,  Fortuitously,  this  between  the  gradient  the  vertical  The  since  of  In  This  A.  quantities  quantities  a  gases  curve  being  as  values.  inconsistent  values  "1".  correlation  coefficient  same  the  length  which  9.43,  the  affecting  characteristic crystal  are  the  al . this  correlation,  14 4 '  case  a  however,  analysed. above  pressure  discussion and  gas  i t  is  composition  234  V e r t i c a l Heat  Transfer  5 id 1  £ u O c cu •o o v.  o  1  1  1  1  Coefficient  (W/cm  10 i d  3  1  1  1  1  1  /  s A  V  1  1  2 00  K)  3  1  1 _  ,  /  ^x  ^^^^  - <r sY  1  1  1  0  1  1  1  1  1 2  I  Horizontal  Heot T r a n s f e r (Arbitrary  Gas  P r e s s u re  1  Coefficient  Units)  Verticol  Horizontal  kg/cm^ Ar N  2  He  Figure  9.43  20 , 5  o  20 , 5  &  20 , 5  O  C o r r e l a t i o n between the measured temperature gradients across the encapsulant and t h e heat transfer c o e f f i c i e n t v a l u e s as a f u n c t i o n o f g a s p r e s s u r e and gas nature. Curve A from Refs. 143-144 using c o n v e c t i o n from a v e r t i c a l w a l l ; and c u r v e B u s i n g c o n v e c t i o n f r o m an h o r i z o n t a l s u r f a c e .  235  may  affect  the  the  heat  not  enough  stress  transfer  changes  to  coefficient  explain  associated of  the  inclusion  the  heat  interface. the  gas  the  this  across  stresses  effect  of  gas  calculations  9.7  The  shapes assumed  p  is  with  of  to  =  and  of  a  a  the  crystal  +  -  interfaces  not  an  full  requires  the  the  heat  smaller  temperature  is  density A  environment  and  reliable not  is  encapsulant-gas  in  flux  to  thermal will  give  profile data  available,  on  in the  precise  Interface  solid-liquid convex  melt. of  the  gas  done.  In  both  revolution  interface  and  concave  cases  the  which  is  on  the  interface  interface  is  represented  by  functions  nondimensional the  and  compositon were  at  the  the  in  2  with  the  the  analytical  are  Such  model  considering to  p  the  non-planar  respectively ;  and  paraboloid  ±0.3  5  in  variables  decreases  Since  and  change  composition.  results  Solid-Liquid  studied  be  which  matter  the  respect  following  £  this  effect  stresses  the  pressure  Curvature  pressure  A  dislocation  and  mechanisms  crystal.  input  crystal on  these  encapsulant.  the  is  on  gas  crystal.  the  effect  of  surface  in  the  pressure  transfer  the  environment  large  gas  in  in  between  effect  decrease  at  gradients lower  A  the  with  description of  distribution  origin signs  respectively.  The  radial  situated  correspond value  of  to the  and  axial  at  the  the  convex  constant  position  axis  0.3  of  and  the  concave  gives  the  236  relative  displacement  interface. for  a  The  convex  manually. with  and  The  the  of  a  the  center  case  i s shown  concave number  shape  is  and  interface. of  of  i n Figure They  growth  considered.  edge  the  9.44(a)  are  generated  conditions  combined  The  most  important  follow.  interface  effect  crystal  in this  (b) f o r a  effect  Convex  The  used  interface  observations  9.7.1  mesh  between  of a convex  length,  thermal  interface  combined  with  and  crystal  conditions  the e f f e c t radius  of is  considered.  9.7.1.1  Crystal  Conditions  :  Length  R,  27.5  B , 21  mm  mm  ;  The length the  the  temperature  of  55  figure  crystal  with  and  crystal  mm  the  isotherms  shape  :  a  giving  the a x i a l  AP,  profile  30  as  i n Figure R,  mm  =  (c)  mm  = 3 R .  and  82.5  Von  temperature  present  ;  ( a ) 27.5  a r e shown  flat  30°  atm. ;  ;  Temperature CL  CA,  Mises  i n Figure field  interface. a  large  negative gradients  are  mm  = 2  fields  for a  9.45.  In  right  different  the  lower  curvature  radial  ( b ) 55.0  curve  stress  is  In  8.19,  than  half  of  following  gradients. larger  the  In  this  at the c r y s t a l  R,  crystal part  that the  the  4,  of  for a crystal  interface  part edge  of the than  237  Figure  9.44  E l e m e n t and nodal c o n f i g u r a t i o n at the i n t e r f a c e f o r c r y s t a l s w i t h c u r v e d i n t e r f a c e , (a) Convex i n t e r f a c e , (b) Concave i n t e r f a c e .  238  at  the axis.  distance  from  The of  edge  field  9.45 i s d i f f e r e n t The  i n the c r y s t a l ,  stress  a planar  about  14  than  less  than  for  the  stress  below  the shoulder.  low  stresses,  h a s moved  with  i n the right  part  observed  discussed  a  values  for a  On  stresses  to  a  i n two  and  at the  and n a t u r e  of the  previously  planar  with  be  shoulder  o f t h e VMS  f o r the case  in this  region i s  interface.  The  stress  i s unique  fora  curved  which  which  planar  to the center  The l o w e s t  is  obtained  a r e 30  t o 40 *  larger  than  the region  with  the other  hand  interface  i s at  and lower  part  are three  times  the interface  of the c r y s t a l smaller  than  near fora  interface.  The the  which  i s shown  the crystal  a t t h e i n t e r f a c e edge, stress  planar  VMS  was  shows  axis.  n o t change  the d i s t r i b u t i o n  i n t e r f a c e . The m a g n i t u d e *  do  i n t e r f a c e . The o r i g i n  interface,  the  which  below  concentration  concentration  edge,  at the axis  largest  of the s o l i d - l i q u i d  first of  stress  interface.  regions  gradients  the interface.  Von M i s e s  Figure  planar  The a x i a l  largest  large  stress  radial  close  interface  to  stress  at the i n t e r f a c e  and a x i a l the  i s associated  gradients  crystal with  edge  i s associated  i n that  axis  and  the nearly  region.  about  constant  2  The  cm  lowest  from  gradient  with  the  in this  region.  The Figure Figure  MRSS  contours  9.46(a-c). 9.46(a),  f o r the three  Comparing i t i s  the  observed  crystal  lengths  right  part  of  that  t h e VMS  a r e shown i n  Figure  9.45 a n d  a n d MRSS  relative  239  Figure  9.45  T e m p e r a t u r e (10 C) a n d V o n M i s e s s t r e s s (MPa) fields for a crystal with a convex interface shape. Radius 27.5 mm. L e n g t h 2 7 . 5 mm. E n c a p s u l a n t t h i c k n e s s 21 mm. T e m p e r a t u r e p r o f i l e a s s h o w n i n F i g u r e 8.19, c u r v e 4.  240  Figure  9.46  MRSS (MPa) contours for a c r y s t a l with convex interf a c e at t h r e e l e n g t h s , ( a ) 27.5 mm, ( b ) 55 mm, (c) 82.5 mm. The largest s t r e s s e s at the interface edge are ( a ) 7.22 MPa, ( b ) 9.19 MPa, ( c ) 9.66 MPa. Radius 27.5 mm, Boron oxide thickness 21mm. Profile as in F i g u r e 8.19, curve 4.  241  distributions  are similar.  approximately  half  The shown is  effect  possible  interface  extends  the interface  this  region.  The  has a  crystal  of  with  flat  of  length  comparable  The  (b) in  crystal to  (c).  axial  are  Figure two  regions stress of  from  with region  the c r y s t a l  with  at  crystal shape.  zero  stress  stresses  appears  shows  radius  of the  length  The  above  and i s  region  above  not extends  that  further  in a the  than  a  the  interface  edge  (b) and s l i g h t l y to a  increases  increases  from  slight  increase  (Yield)  i s shown  gradient.  the lowest also  part  i t i s concluded  i s attributed  f o r the three  with  a  radius.  (a) to  effect  to the  to the d i s t r i b u t i o n  this  does  close  and  i s the crystal  d i s t r i b u t i o n s o f t h e MRSS-CRSS  9.47(a-c) regions  value  region  i n the lower  similar  From  specific  of approximately  interface  interface  stress  temperature  stress  change  interface.  last  the  is  distributionsi t  with  The s e c o n d  to the c r y s t a l  This  i s  point.  distribution  the three  distance  distribution  length  t h e MRSS  distribution,  little  convex  largest  The in  this  to a  f o r t h e MRSS a r e  a t any  regions  One  the convex  stress  effect  with  stress  in  two  at the axis.  experiences  characteristic  levels VMS  Comparing  distinguish  from  this  length  characteristics.  which  crystal,  crystal  9.46(a-c).  to  reproducible  absolute  of the corresponding  of  i n Figure  The  crystal which from  i n (b) and  a large  area  with  lengths.  In (a)  are associated Figure  9.46(a).  with  there the  This  zero  (c).  In ( c ) t h e upper  part  zero  stress.  242  9.47  MRSS-CRSS ( Y i e l d ) (MPa) contours for a crystal with convex i n t e r f a c e at t h r e e l e n g t h s ( a ) 27.5 mm, (b) 55 mm, ( c ) 82.5 mm. Radius 27.5 mm, boron oxide t h i c k n e s s 21 mm. P r o f i l e a s i n F i g u r e 8.19, curve 4.  243  The  dislocation  depends  on  (Yield)  i n Figure  the  the  position  plane  from  MRSS-CRSS centre,  ends  maximum less  is  is half the  than  results  3.0  MPa  occurs  tail  shaped.  The edge  The depends Close  on  maximum  of the U  of  This  to  the  the  i s three  the of  times  MRSS-CRSS  cross-section  i s at a distance  stress  i s reached  i s 2 7 . 5 mm at this  bottom  stress  a  W-  diameter  than  the  in  (001)  (Figure  at  cross-  have  to  the  Similar  Thus  has t h e symmetry  long  point.  and  i s always  respect  (Yield)  the c r y s t a l  the  except  distribution  t h e MRSS-CRSS  when  from  stress  a  higher  with  the  t h e edge.  (Yield)  the plane  At  reach  will  (Yield)  with  never  interface.  cone  crystal  cone.  t h e maximum  d i s t r i b u t i o n along  t h e MRSS-CRSS  the p o s i t i o n  9.48.  cone  end where  at  the  t h e cone t h e  radius  and  e n d , t h e maximum  at the t a i l  (Yield)  the  of a  low  the value  where  symmetry  t o t h e cone  Figure the  end  are  the  At the seed  i s ,half  from  MRSS-CRSS  the  stresses  distance  shaped  distance  after  increasing from  to  a t the edge.  close  a t any  close  at a  very  sections  at  that  the seed  that  are obtained  centre  double  that  as w i t h  near  o f 50 mm  MRSS-CRSS  right  surface  gradually  growth  suggests  a r e formed  the cone,  At a d i s t a n c e  the  At  surface  large  edge  (Yield)  stresses  crystal  The  MRSS-CRSS  the s t r e s s  stresses.  crystal.  the encapsulant  to  t h e cone.  a t t h e end o f  at the i n t e r f a c e  axial  close  the largest  of  the  the c r y s t a l  The  (Yield)  interface  in  and not above  i s lowest  distance  the  at  interface.  surface  i n the c r y s t a l  9.47(a-c)  dislocations  solidification  density  i s U-  centre.  planes  the  cone.  shown i n  o f 1 1 . 0 mm 9.47(a)  from  ). The  244  Figure  9.48  MRSS-CRSS ( Y i e l d ) (MPa) contours i n a (001) p l a n e at a distance of 11 mm from the cone i n the crystal shown i n F i g u r e 9.47(a).  245  The more  symmetry  than  growth  55  will  o f t h e MRSS-CRSS  mm  f a r from  t h e ends  be  determined  by  shown  i n Figure  9.49(a-c).  Despite  were  taken  observed same  from  that  patterns  (a) )  along  a  this  region  there  dislocations  9.7.1.2  from  as shown  Effect  Conditions  and s t r e s s  are  results  shown  9.51(a-b)  for a  corresponds CRSS  to  (Yield)  The  12.5°C/cm  MRSS  pass  through the  to the i n t e r f a c e an U - d i s t r i b u t i o n  symmetry  i s present.  to a  have  the  symmetry  and  In same  of the  Gradient  as i n F i g u r e  distributions  for a  50°C/cm  gradient.  distribution  distributions of  shoulder.  stress  The  below  i t was  f o r two gradient  Part and  9.35(b),  (a) part  in  gradients and  each  (b) t o  Figure figure  t h e MRSS-  distribution.  concentration  than  9.47(c)),  with  show  Thickness  profiles  9.50(a-b)  t h e MRSS  sections  9.49(a).  f o r the s t r e s s  i n Figure  end  changes  these  Close  expected  tail  a t t h e end o f  symmetry  t h e ends  levels.  which a r e  ;  Temperature  The  (Figure  are  i n Figure  B, 50 mm  that  eight-fold  the  of Encapsulant  :  the fact  symmetry  dislocations  Sections  of  f a r from  A t t h e edge  the  sequence  crystal  i s four-fold  diameter.  symmetry.  one  in sections  of the c r y s t a l  the  a l l the sections  symmetry  (part  only  (Yield)  stress  at  for the  the  interface  at the interface  the shoulder.  two  The lowest  edge  gradients edge  and  i s about  stress  level  show below  80 % occurs  a the  larger at the  246  a  247  248  Figure  9.49  MRSS-CRSS ( Y i e l d ) (MPa) c o n t o u r s in (001) p l a n e s at t h r e e d i s t a n c e s form the cone. ( a ) 7 9 . 7 5 mm, (b) 6 6 . 0 mm, ( c ) 5 2 . 2 5 mm. S e c t i o n corresponds to t h e c r y s t a l shown i n F i g u r e 9.47(c).  249  Figure  9.50  ( a ) MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) (MPa) f o r a crystal with convex interface. Boron oxide thickness 50 mm. Radius 2 7 . 5 mm. Length 55 mm. A v e r a g e g r a d i e n t 50°c/cm.  250  Figure  9.51  (a) MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) (MPa) for a crystal with convex interface. Boron oxide thickness 50 mm. Radius 27.5 mm. ' Length 55 mm. Average gradient 17°C/cm.  251  centre edge  of  the c r y s t a l .  decreases  The  shows  substantially large  areas  larger  9.7.1.3  (Yield)  zero  to  t h e CRSS  Effect  Conditions  :  shown  MRSS  in  Figure  the  CL,  crystal  two  common  The to  the  largest note  that  interface  stress while  as t h e r a d i u s  interface  edge  symmetry described  of  to  stresses  area  show  f o r t h e two  of  For  ;  B,  this  both  region  conditions  stresses  and  (Yield) (b)  which are  ;  8.19,  for this  obtained  (001) p l a n e s  27.5  stress  It i s  show  On  surface. important  the  same  surface  level  the other  mm  levels  the encapsulant  %.  are  stress  encapsulant edge.  4.  crystal  the  (45 % ) , t h e s t r e s s 10  curve  The  with  Large  the  above  about  crystal.  mm  respectively.  above  level  21  i n Figure  at the i n t e r f a c e  increases  f o r t h e 2 7 . 5 mm  a  those  increases  in  mm  characteristics.  the s t r e s s  increases  80  and  edge  occurs  only  edge  profile  9.52(a) similar  at  The  gradient.  MRSS-CRSS  are  exist  gradient.  Radius  ;  distributions with  in  interface  (Yield).  40 mm  and  the  the i n t e r f a c e  Temperature  The  at the  (b) of the f i g u r e s  regions.  with  value  a reduction  i n part  of c r y s t a l  R,  stress  with  stress  increases close  than  largest  proportionally  MRSS-CRSS  gradients  The  at the  hand  the  features  as  252  253  Concave  9.7.2  Interface  Conditions  B,  21  mm  Temperature  The radii 27.5  27.5  b)  R,  40  in  the  may  be  low  for  concave  interface  In  the  another  shape.  largest  second is  The  stresses  largest  dislocation the  concave  similar  density  in  the  centre  15  *  The is  effect  different  of  than  to  is  that  increasing that  the  the  obtained  the  to  are  of  above  a  stress  follows  the  at  the  W  stress  interface.  therefore  the  distribution  at  shape, the  of  the  planar and  are  crystal.  interface,  with  this  which  present  the  the  comparable  characteristic  the  the  stress  close  region  interface  direction  than  One  interface  this  for  the  distributions  obtained  by  crystal  (a)  interface  centre  the  two  Part  the  are  above  at  For  the  stresses  the  controlled  radial  higher  at  the  interface,  by  large  4,  radius.  from  and  well  occur  interface.  distribution about  stress  (b).  mm  stress  Very  for  regions.  formed  radii  region,  distribution  two  has  crystal  and 40  distance  region  both  with  a  curve  mm.  convex  in  8.19,  mm  (Yield)  the  the  divided  and  similar  interface  of  Figure  55  9.53(a) for  in  80  MRSS-CRSS  (b)  extending  The  CL,  Figure  case  and  mm  part  as  CL,  mm  the  the  radius  section.  in  and  distribution interface  for  shown  radius  As  to  R,  results  are mm  a)  profile  the  shape  stress with  the  edges.  radius for  for a  a  convex  concave  interface  interface.  For  254  Figure  9.53  (a) MRSS (MP) and ( b ) MRSS-CRSS ( Y i e l d ) (MPa) a c r y s t a l with concave i n t e r f a c e , ( a ) R a d i u s 27.5 ( b ) 40 mm. B o r o n o x i d e t h i c k n e s s 21 mm.  for mm,  255  the  concave  interface  the  largest  stress  The interface to  a  value  from  symmetry  of  i s shown  in Figure  section  crystal.  increasing  at  Similar  2.75  the  mm  symmetry  the  14.26  MPa  9.54.  from  central  region  eight-fold  fold  symmetry  i s also  observed  this  case  eight-fold  at  (001)  a  ring  t h e edge  for mm  increases  (30 * ) .  to  to  the  corresponds the  27.5  crystal.  i s observed.  close  i s not  %  close  plane  f o r t h e 40  symmetry in  MPa  interface  i s observed  45  formed  This  the  about  t o 18.86  dislocations  the  typical  radius  the  observed.  mm In  Four-  edge.  In  256  Figure  9.54  MRSS-CRSS ( Y i e l d ) (MPa) contours i n a (001) p l a n e a t a distance of 2.75 mm from the interface in the c r y s t a l shown i n F i g u r e 9.53(a).  257  CHAPTER  RESULTS  Using in  Chapter  stress are  the  obtained  110.0  mm.  combined  7  and  fields  This  a  and  axial  temperature  analytical  cooling  crystal  length  stress  COOLING  evaluated  during for  for  and  the the  GROWTH  numerical  Chapter  after  with  AFTER  a  radius  27.5  the  end  scheme  the  are  to  calculated. mm  made  and  with  a  Results  for  of  which  growth.  is  the  and  length  length  during  growth  model  described  temperature  largest  of  cooling  8  growth  c a l c u l a t i o n s are  at  field  in  corresponds  temperature  temperature  FOR  10  The  the  initial  radial  gradient  neglected.  In crystal the  the  calculation  after  and the  calculations, R  above  the  designated the  as  gradients  The  800  and  1000°C .  thermal  crystal  are  GC ,  GC/2  i n the  or  1000°C  as  boron and  are and  GC/4.  the  characteristics major  These  i n the 70,  during  boron  argon oxide  the  region  35  in  the  profile  of  gradients  i s surrounded  The  fields  significance.  gradients  chamber  i t cools,  stress  temperature  considered,  growing  oxide.  and  initial  of  temperature  interface  thermal  the  the  three  crystal  atmospheres  the  solidification,  crystal,  surrounding  of  and  in  media In  of  the  length  17.5°C/cm  are  about  half  solidification.  by  either  temperatures temperature  argon  at  considered is  taken  30 are as  258  10.1  Ambient  10.1.1.  Temperature  Argon  10.1.1.1.  Initial  The  Gradient  temperature  vertical  section  is  in Figure  shown  about  1 cm  rapidly. to  The  at  The  becomes  be  lower  negative of  calculated become  shown  of  the  axial  in  than  the  are  the  crystal  gradients  temperature levels  in  which as  are  the  The  case,  throughout  and  a  the  positive  gradient  isotherm, a  crystal.  the  model low.  and  changing  the  of  temperature  ambient  ambient  a  dropped  from  is then  ambient  temperatures  temperatures  too  of  region  has  radial  With  surface  in  changes  1000°C  temperature. the  that  high,  80°C.  and  half  temperature  region  about  right  from s o l i d i f i c a t i o n  show  the  this  normally  significant  the  seconds  results  gradients  lower is  5  interface,  interface  temperature  is  less  the  over  after  The  large,  at  stress  70°C/cm  resulting  The  will This  in  assumption likely  give  error  will  temperature  gradients  smal1er.  The  radial  is  ambient  less  become  to  which  radial  constant  crystal  the  positive  temperature, will  a  gradient  edge  =  distribution  radial  between  GC  10.1(a).  axial  the  negative  of  adjacent  negative.  drop  1000°C  in  Figure  distribution 10.1(b).  near  gradients.  In  (b)  the  and  the  distance  in  gradient  (a)  changes  interface  sign  seconds  Comparing  isotherms  than  the  10  axial  has  (a) has  after and  (b)  increased  gradient above  solidification  the  increased.  at  the  the  curvature  giving  higher  interface  interface  where  is the  260  c Figure  10.1  d  Temperature (10 C) field during cooling at four times. ( a ) 5 s, ( b ) 10 s, ( c ) 20 s ( d ) 60 s, Initial gradient c l o s e t o i n t e r f a c e 70 C/cm. Argon t e m p e r a t u r e 1000 C 3 o  f a  261  The  temperature  10.1(c))  shows  interface seconds much  continuing  and a  10.1(d))  uniform.  temperature  high  considerably  (Figure  more  distribution  By  after  radial  axial  the temperature seconds,  distribution  is  seconds  gradients  decreasing  300  20  almost  close  to  gradient.  distribution  the  (Figure  After  has  calculations  uniform  the 60  become  show  the  throughout  the  crystal.  The  very  rapid  solidification assumed during  in  shows  crystal  10.1.  other larger  than  The  Heat  heat  interface.  A  the  five  minutes  state  after  approximation  temperature  midway moves  move  stress between  radial  distribution  Figure  10.1(a-d)  process  occurs  the observation  considerably at  from  the  i n the  i n the bulk  a  that  to  surface  would  crystal.  crystal  d in  On  would  the give  gradient.  determined  Figure  region  from  lO.l(a-d)  stresses  the  from  upwards  control  in  transfer  control  isotherm  in  heat  concluded  (Yield)  towards  isotherms  the  transfer  largest  low  the  not  transfer  shown  The  which  does  such  MRSS-CRSS  10.2(a-d).  axis  of  of  c a n be  the observed  distributions  interface  This  move  hand,  of  control  isotherm  considerably  first  growth.  cooling.  Figure  the  quasi-steady  calculations  mixed  1000°C  in  the  examination  that  during the  supports  the  Further  cooling  are  outside  the axis  with  are  observed  i s present  temperature  shown to  about  surface time.  the  in  occur 1  and  cm  Figure at  the  above  the  the  As a r e s u l t  vertical of the  263  c Figure  10.2  d  MRSS-CRSS ( Y i e l d ) (MPa) d u r i n g c o o l i n g f o r the four temperature fields shown i n F i g u r e lO.l(a-d). (a) 5 s, (b) 10 s, ( c ) 20 s, ( d ) 60 s. Initial g r a d i e n t 7 0 ° C / c m . A r g o n t e m p e r a t u r e 1000 C.  264  movement  the  changes  from  The about 4.5  when  3  growth  increase crystal  stress  is  10.3(a-d), observed  1)  2)  the  main  and  the  This  density  crystal  the  interface  ranges  appreciably  i n the c r y s t a l  diameter  would  between larger  during  crystal  above  result  the  18  than  growth  in  i n the v i c i n i t y  occurs  the  and a r e  boron  a  and  oxide  three-fold  o f t h e end o f t h e  cooling.  (Yield)  27.5  mm  for crystal  from  the  cross-sections  interface  characteristics  of  are  the  at  shown  in  stress  2.75, Figure  symmetries  are :  At  2.75  mm  from  the  eight-fold  symmetry  a  is U  diameter  A t 8.25  mm  symmetry  3)  At  <100>  At  from  16.5  mm  There  interface  and  the  the s t r e s s  outer  ring  has  distribution  an  across  shaped.  the i n t e r f a c e  and t w o - f o l d  symmetry.  4)  are  during  and  near  values  in  the  time.  stops,  stopped.  MRSS-CRSS  16.5  with  growth  stress  in dislocation  along  (Yield)  position  the  generated  The 8.25,  after  t h e same  times  shape  MRSS-CRSS  These  at  distribution  to a U  seconds  MPa.  about  a W  largest  10  values  stress  from  the centre  has  four-fold  symmetry..  the  are four  interface minimum  there  stress  is  circular  regions  i n the  directions.  27.5  symmetry.  mm  from  the  interface  there  is  two-fold  265  266  267  C  268  Figure  10.3  MRSS-CRSS ( Y i e l d ) (MPa) contours i n (001) planes at four distances from the bottom (interface) in the crystal shown i n F i g u r e 1 0 . 2 ( b ) 10 s, ( a ) 2.75 mm. ( b ) 8.25 mm ( c ) 16.5 mm ( d ) 27.5 mm .  269  The 2.75  stress  mm  from  10.3(a)  at  symmetry the  10  and  change  the  at  in  symmetry  the  changes  Figure  four-fold 60  and  interface  seconds  shown  two  distribution  eight-fold  at  60  10.4(a).  to  the  in  time  8.25  mm  Figure  two-fold  dependent.  symmetry  seconds  At  symmetries  seconds  are  to from  Figure  the  eight-fold  the  interface,  10.3(b)  symmetry  in  At  at  10  shown  seconds  in  Figure  10.4(b).  10.1.1.2  In  Initial  this  case  temperature  it  is  have  larger  CRSS  (Yield)  the  in  seconds  MRSS-CRSS  interface strong  of  has  directions  to  a  symmetry as  distributions  shown along  a  the  in  a  in  At  70°C/cm  by  weak  Figure is  8.25  10.7(b)).  circular  diameter  35°C/cm  is  stresses  10.5(b). a  gradient.  cross-section  (Figure  given  for  70°C/cm gradient  The  similar  MRSSto  gradient.  are  the  still  the Large  present  10.6(a-b).  symmetry.  near  10.4(a) the  The  and  Figure  10.5(b)  large  seconds.  in  for  the  10  10.5(a)  70°C/cm  for  Figure  (Yield)  field  Figure  and  at  Figure  Figure  for in  in  maximum  isotherms  10.4(b)  symmetry  changes  four-fold  the  than  eight-fold  two-fold  gradually  that  shown  in  with  isotherms  as  a  (Yield)  10.5(a)  Figure  35°C/cm  reach  shown  distribution  curvature  The  is  curvature  in  =  stresses  noted  distribution  60  GC/2  MRSS-CRSS  Figure  gradient  at  the  field  corresponding Comparing  Gradient  at  mm The  symmetry local  10.7(b). present  the  mm  from  stress  two-fold with  minima The  in  2.75  has  a  symmetry  small in  the  the  U-shaped  traces <110> stress  a l l cross-sections.  270  271  Figure  10.4  MRSS-CRSS ( Y i e l d ) (MPa) in (001) planes at distances from the bottom i n the crystal shown F i g u r e 10.2 ( d ) 60 s, ( a ) 2.75 mm ( b ) 8.25 mm.  two in  272  b Figure  10.5  (a) T e m p e r a t u r e field (10 C) and ( b ) MRSS-CRSS (Yield ) (MPa) field for a crystal c o o l i n g i n argon a t 1000 C a f t e r 10 s . I n i t i a l g r a d i e n t 35°C/cm.  273  Figure  10.6  (a) Temperature field (10°°C) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d for a crystal c o o l i n g i n argon a t 1000 C a f t e r 60 s . I n i t i a l g r a d i e n t 35°C/cm.  274  a  275  Figure  10.7  MRSS-CRSS ( Y i e l d ) (MPa) in (001) planes at distances from the bottom i n the crystal shown F i g u r e 10.5 ( b ) 10 s . ( a ) 8.25 mm ( b ) 27.5 mm.  two in  276  10.1.1.3  The have  Initial  temperature  the  The  close  region  adjacent the  The  to the  The  (Figure  10.8(a))  among  (Yield)  interface  large  (Figure  near  isotherms  initial  10.8(b) to  the  i s expanded  stress  gives  the  is similar  stress  to the i n t e r f a c e  crystal.  17.5°c/cm  curvature  MRSS-CRSS  with  =  GC/4  field  highest  considered. region  Gradient  i n the  the previous  cases.  crystal  along  gradients  field  surface  t o encompass  distribution  which  the  nearly  and  a l l of  diameter  i s U-  shaped.  The  MRSS-CRSS  (Yield)  in cross-sections  1)  e i g h t - f o l d symmetry  close  2)  two-fold  above  symmetry  distance  3)  Near with the  with  larger  large than  distribution  These  area  give  persist  ones  distribution  tends  o f 2 7 . 5 mm  10 to  a  o f 16.5  a t 2 7 . 5 mm  and  up t o a  mm  from  covering  the i n t e r f a c e more  than  half  and l a r g e  stress  gradients  U-shaped  stress  distribution  at  seconds. take  from  after  observed  in cross-sections at  interface  at  base.  stresses the  the  i n the centre  surface  a broad  distribution  distance  symmetry  low s t r e s s e s  edge.  to the i n t e r f a c e  the i n t e r f a c e  circular  section  the  The  from  shows  a  V  at 60  Along shape  the i n t e r f a c e  60  seconds 10  seconds.  seconds a  but they The  stress  is similar  to the  diameter  rather  are not  than  the central  the U  shape.  region  stress At  a  i n the  277  Figure  10.8  (a) T e m p e r a t u r e field (10 C) and ( b ) MRSS-CRSS f o r a c r y s t a l c o o l i n g i n argon (Yield ) (MPa) f i e l d gradient 17.5°C/cm. a t 1000 C a f t e r 10 s . I n i t i a l O U  278  wafer  shows  10.9). 10  a  The  and  dislocation  fields  in  10  a  the  GC, and  seconds  The  GC/2 b  show  MRSS-CRSS  atmosphere,  sections  at  5  interface  instantly four-fold  but  (Figure  higher  be  two  but  is  than  at  evident  in  the  initial  thermal  to  and  the  evolve  in At  that  a  that  in  three 10.10  order.  the highest  10  * higher  and  10.10(b)  mm.  27.5  mm  above  do  some  from  near  symmetry  circular  with  present  addition  eight-fold  to  i n magnitude.  In  the  symmetry  obtained  a  16.5  at  stresses.  by  at  10.12.  Results  Figure  replaced  to  and  thermal to  cooling  distributions  seconds  is  close  respectively.  the  cross-sections  two-fold  from  17.5°C/cm  in Figure  a r e 20  present show  for  in  the  stresses  and  oxide.  distributions  symmetry  also  seconds  symmetry  fields  (Yield)  two-fold  shown  exhibits  i n shape  same  70°C/cm  fields  are  characteristics.  which  the  GC/4  which  MRSS-CRSS  symmetry  may  i s boron  stress  the  similar  the  is  f o r the  above  medium  (Yield)  an  interface  repeated  and  a r e shown  distinctive  section  symmetry  used  and  are  The  the  are  crystal  10.12(b) argon  this  symmetry  Oxide  temperature  gradients  eight-fold  distribution.  the s u r r o u n d i n g  The  in  therefore  calculations  assuming  Parts  level  density  Boron  The  distinguished  stress  seconds  10.1.2.  well  the  circular crossclose not  symmetry  to  form and  a  279  Figure  10.9  MRSS-CRSS ( Y i e l d ) (MPa) i n (001) planes a o f 27.5 mm from the bottom i n the c r y s t a l F i g u r e 10.8 ( d ) 10 s .  distance shown i n  0  280  a Figure  10.10  b  (a) Temperature field (10 C) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n boron oxide at 1000°C after 10 s. Initial gradient 70 C/cm.  281  Figure  1 0 . 1 1 (a) Temperature field (10°"c) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d for a crystal c o o l i n g i n boron oxide at 1000°C after 10 s. Initial gradient 35°C/cm.  282  b Figure  10.12  (a) T e m p e r a t u r e field (10 C) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d f o r a c r y s t a l c o o l i n g i n boron oxide at 1000°C after 10 s. Initial gradient 17.5°C/cm.  283  10.2  Ambient  In the  Temperature  this  boron  case  oxide  The  only  layer  initial  The  stress  when  surrounding  the  in  most  1000°C,  the  atmosphere  and  the  media  are are  was  after  exhibit  difference  being  10  seconds  be  the  those  oxide.  of  to  to  that  obtained  The  different  lack  three  10.13  similar  between  few  for  Figures  boron  a  since  800°C.  to  levels and  considered  in  seen  argon  is  above  shown  stress  cross-sections  prominent  well  fields  distributions  for  fields  stress  gradients  obtained the  argon  generally  and  temperature  10.15.  an  is  temperature  8QQ°C/CIB  stress  features,  local  minima  in  the<100>directions.  10.3  Analysis  The the  most  last  than  in  smaller very  produce  in  entire  sense,  is  axial  from  levels  importance the  tail  cooling  can  such  the  a l l conditions  in  cooling  to  of  crystal  and  end  in  leading  to  directions.  the argon  boron  value  shown  of  the  will  thermal  in  some  produce  result,  transfer  This  minimize  to  with  in  larger  growth.  This  correlated heat  larger  is  oxide.  effect  larger  much  proper  larger  a  gave  of  since  that  examined  described  during  directly  The  results  developed  be  stresses.  efficiently,  Cooling  crystal. Cooling  than  larger  and  the  for  result  stress  coefficient,  coefficient  radial  the  stresses  broad  Results  that  density  the  transfer  more  is  indicates  dislocation cases  the  significant  section  stresses clearly  of  the  in  a  heat  coefficients heat  cool  the  gradients  transfer surface in  both  284  Figure  10.13  (a) Temperature field (10 C) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d for a crystal c o o l i n g i n argon a t 8 0 0 ° C a f t e r 10 s . I n i t i a l g r a d i e n t 70 C/cm.  285  Figure  10.14  (a) Temperature field (lO^c) and ( b ) MRSS-CRSS (Yield ) (MPa) f i e l d for a crystal c o o l i n g i n argon a t 8 0 0 ° C a f t e r 10 s . I n i t i a l g r a d i e n t 35°C/cm.  286  Figure  10.15  (a) T e m p e r a t u r e field (10 C) and ( b ) MRSS-CRSS (Yield ) (MPa) field for a crystal c o o l i n g i n argon a t 8 0 0 ° C a f t e r 10 s . I n i t i a l g r a d i e n t 17.5 C/cm. O U  287  In the  addition  crystal  thermal of  the  crystal the  is  two  10.17  to  the  in  the  larger  gradient  in  curve  radial  gradient.  from create to  The thermal  In  stresses  in  in  Figure  stress  for  accounted case  radial  the  observed.  constrains  for  a  (a)  GC  both  the  by  largest  radial for  is  (curvature)  thermal  GC/2.  larger  difference  in  case In  is attributed  larger  GC in  axial  gradient  the  condition  changes  and  each  (b)  and  larger  stress  cannot  separately.  In  cases  account  a  gradients.  temperature  and  (a)  Low  surface  interface  temperature.  initial  free  the  radial  argon  can  The  along  the  in  considered.  800°C  curve  gradient  be  gradients  in  for  conditions  axial  10.17  10.16  be  near  considered,  Figure  this  interface  more  higher  plane  change  the  In  a  also  large  oxide  gradient  in  cannot  for  boron  comparable  (b).  the  and  the  stresses  stresses  10.16  are  largest  10.17  gives  radial  1000°C  than  The  rapid  a  must  of  the  of  radial  Figure  the  result of  conditions,  cooling  crystal  Figure  curve  (b)  stresses.  a  gradients  10.16  of  the  terms  for  initial  Figure  in  in  external  start  as  in  shown  Figure  the  magnitude  explained  This  the  gradients  However, be  at  to  (a)  in  axial  or  attributed  to  with  distance  radial  gradient  expansion  which  leads  stresses.  above  analysis  stresses strain  temperature  using  shows  the  approximation  variations.  the  advantage  axisymmetric which  of  calculating  approximation  cannot  account  over for  the axial  200  "  1120-  1080  -  1040  -  Figure  10.16  Temperature p r o f i l e s i n the c r y s t a l c o o l i n g i n argon a t 800 C f o r two initial gradients i n the crystal, ( a ) 70 C/cm ( b ) 35 C/cm. The r i g h t p a r t corresponds to the radial profiles and the left part to the axial profile. Full and broken lines correspond to the a x i s and s u r f a c e t e m p e r a t u r e s respectively.  Figure  10.17  Temperature p r o f i l e s i n the c r y s t a l c o o l i n g i n boron oxide at 1000°C f o r two initial gradients i n the crystal. (a) 70°C/cm ( b ) 35 C/cm. The r i g h t part c o r r e s p o n d s to the r a d i a l p r o f i l e s and t h e l e f t part to the axial profile. Full and broken lines correspond to the axis and surface temperatures respectively.  to oo  290  CHAPTER  SUMMARY  In  the  crystals  present  has  been  during  were  assuming  that  conditions  differential  equations  techniques.  For  state  conduction  equations  were  conditions interface Linear  the  triangular  crystal.  The  obtained  using  quadratic Numerical  the  values  analytical  numerical  elements  and  analytical the  for  the  energy to  finite  element were  solutions. numerical  the  for  compared  the  method  solid/liquid surface.  discretize  and  crystal  solutions  the were  quasidomain.  satisfying  with  stress  numerical  calculations,  and  plane  were  boundary  calculations  the  addition,  element  The  Linear  radial  with In  and  compared  In  obtained  quasi-steady  to  stress  the  element  crystal  used  discretize  of  that  the  the  the  principle.  and  finite  method. at  at  were  solutions.  fields  with  cooling  conditions  were  axisymmetric  temperature of  used  solutions  temperature  calculated  Galerkin's  law  employed  GaAs  calculations  solutions  employing  using  the  boundary  numerical  in  dislocation  The  calculations  the  to  thermoelastic,  was  were  simplified of  The  equations  of  is  obtained  minimum  elements  related  equation  toroidal  element  values  apply.  distribution  cooling.  material  constant  Newton's  and  and  temperature  assumed and  the  were  obtained  stress  analysed  solidification  axisymmetric  heat  CONCLUSIONS  I n v e s t i g a t i o n the  calculated,  formation made  AND  11  axisymmetric strain  the  and  temperatures  compared  with  291  temperatures model  for  basis  of  measured  the  temperature  these  conclusions  The  finite  results  good  solutions  consisting  of  constant  cm  The  the  grower.  fields  most  were  The  mathematical  evaluated  important  e v a l u a t i o n are  agreement  analytical  0.6  stress  solutions  solutions,  interface.  pressure  on  the  results  given  and  below.  Field  analytical  and  and  from  element  in  high  results.  obtained  Temperature  1.  in a  The The  with  obtained  the  was  the  same  boundary at  as  a  values  problem.  the  end at  of the  used  cylinder  of  gave  of  the  For  the  c o n d i t i o n s were  coefficients  taken  equation  numerical  coefficient  transfer  crystal  temperature  temperatures  transfer  heat  the  for  simplified  constant heat  of  the  used  crystal  crystal-gas were  0.3  aspect  and ratio  two .  2.  Refinement not  of  change  the the  mesh  in  the  numerical  finite values  element of  the  solutions  did  calculated  temperatures.  3.  The  calculated  reported Melbourn  temperatures  temperature crystal  grower  are  measurements at  3.04  MPa  in in  good GaAs  pressure.  agreement crystals  with in  a  292  Stress  4.  The  Field  stress  converge values  5.  For  components  to  different  (nodal  linear  element,  components  6.  For  a  7.  8.  in  strain  close  to  order  a  This  is  on  the  the  element  method  initial  strain  calculations.  displacements in  the  of  configuration,  constant  converge  finite  inside  element  magnitude  gives  faster  attributed  the  pseudo- q u a d r a t i c  For  radial  initial than  to  the  strain  with  the  stress  higher  than  use  the of  linear gives  element  stress  formul-  components  pseudo-quadratic an  incomplete  B  element. matrix  in  using  fields,  the  average  stresses  calculated  by  temperatures  are  in  good agree  agreement  with  analytical  axisymmetric  solutions.  They  less  with  analytical  plane  solutions.  The  element  solutions  markedly  from  For with the The with  the  axisymmetric the  finite  stresses  the  using anlytical  element  finite  element  initial  strains  finite differ  solutions.  fields  method using  axisymmetric  strain  linear  temperature  calculated  analytical  a  element.  temperature  element  well  the  strain.  nodal  that  finite  used  initial one  the  depending  quasi-quadratic  initial  with  by  linear  given  ation  values  temperatures)  and  a  constant  calculated  are  the  stresses  three  times  plane  stress  solutions  the  strain  values due  to  calculated smaller  approximation.  agree the  than  less  divergency  well of  293  the in  Bessel  function  the Fourier  Specific  20  mm,  consist  9.  cone  series  results  Calculations  were angle  RSS  a r e added  values  gives  stress  times  larger  concept  Changing  values  model  formulation  were  length  10  crystal  than  the  surrounding  10  stress  local  terms  mm.  obtained. mm,  radius  The  results  crystal  times  larger  than  3  a in  axial  gives  proportionally  environment  except  the  The to  in  the  used  decreases  i n a span  strong  except  gradient.  the  i s inconsistent  in  gradient  the environment,  environment  are  7  VMS with  representing  the  crystal.  The  20 %  distributions  which  RSS  not s u b s t a n t i a l l y change the  gradient  400°C/cm  t h e 12  t h e 12  which  components  does  has  in  Adding  levels  distribution  only  i n which  stress  gradient  of  RSS.  coefficient  temperature  the  reported  temperature  the  of  quantity.  values  stress  change  and  t h e RSS  transfer  and  a with  t h e MRSS  Adding  the heat  been  to give  of a t e n s o r i a l  levels  Changing  of  of  thickness  have  distributions  % of the o r i g i n a l  stress  and  numbers  of the following.  temperature  11.  f o r large  i n the s o l u t i o n s .  45°, encapsulant  stress  53  the  for a  Calculated  the  order  employed  on  made  distributions.  10.  of f i r s t  crystal  effect  environment  on  crystal. is  at  temperature The  usually  stress  the decrease gradients  one  axial  half  the highest  gradients  maximum  at the lowest  the  the  gradient  quarter  of  i n the  crystal  i n gradient  of the  near  50°c/cm.  294  12.  A  stress  concentration  shoulder  for  thickness.  crystal  The  usually  lengths  occurs  below  comparable  concentration  to  disappears  the  the  at  crystal  encapsulant  low  gradients  (50°C/cm).  13.  14.  Increasing  the  gradient  effect  the  operative  to  system.  The  slip  symmetries  in  The  field  transverse  stress  usually  exhibits  symmetry  is  an  Increasing  associated  the  transverse  eight-fold with  encapsulant the  the  and  stress  The  temperature  are  sections  of  slip  has  little  on  stress  changes  planes  symmetry the  400°C/cm  at  mode  minimal.  the  the of  crystal  edge.  the  The  MRSS  which  distribution.  changing  surrounding  ly  on  eight-fold  without  16.  100°C/cm  distribution  has  15.  on  from  thickness  temperature  crystal  has  from  10  mm  to  25  mm  profile  in  the  environment  effect  on  the  temperature  little  distribution.  influences  profile  the  shape  stress  level  in  the  and  boron  stress  oxide  significant-  distribution  in  the  gives  the  changes  the  crystal.  a)  A  constant  lowest  b)  A  stress  gradient  symmetry,  transverse  across  the  encapsulant  values.  non-constant  stress on  gradient  resulting  planes.  in in  the  boron  stronger  oxide  four-fold  symmetry  295  The  model  condition.  The main  Cone  The  17.  cone  to  results  angles  study  the  effect  and c o n c l u s i o n s  considered  thickness  20 mm.  The  used  of  the  are given  growth  below.  Angle  Encapsulant radius  was  VMS  and  The r e s u l t s  and  MRSS  were  7 ° , 30°, 4 5 ° , 54.7°  crystal  indicate  length that  distributions  were  10  and  mm  65°.  crystal  :  are  similar  f o r each  cone  angle.  18.  T h e VMS 54.7°  exhibits which  plane.  due  low for  gradients.  by  T h e MRSS  levels  20.  T h e MRSS  gives  angle  cone plane  i s due  generally i s two  to  For a  The s t r e s s  symmetry  angle.  2.5  central  area  parallel  stress  decrease angles  for  in  mm  in from  changes  to  radial  the s t r e s s  angle  the (111) can  be  temperature  increases  possibly  field.  half  times  surface  angle.  t h e VMS larger  the condition cone  t h e cone  behaviour  a r e i n d e p e n d e n t o f t h e cone  t h e RSS c o m p o n e n t s  At  level  cone  the  i n the s t r a i n  levels  surface.  a  angles  cone  t h e MRSS  singularity  stress  to  the  At l a r g e  19.  21.  cone  to constraints  cone  minimum  corresponds  At  accounted  a  of  levels.  than no  At  t h e VMS.  traction  coinciding  54.7°  with  The  at the a  (111)  a r e a t a maximum.  transverse the  from  planes  interface  eight-fold  the  changes  with  symmetry  f o r 7°  cone  cone  i n the angle  to  296  four-fold from  for  the  the  7°  the  edge  For  the  interface  cone  in  are  same  22.  for  any  the to  were a  21  studied mm  24.  For is  a a  the  always these  critical  sections  from  larger  At  symmetry. may  because  mm for  angles.  symmetries  yield  50  four-fold  eight-fold  distributions  27.5  for  length  not  the  be  stress  stresses.  The  surfaces  up  to  40.0  110  mm.  i t is  mm  long.  The  Two  conclusions  specified  that  the  conditions.  radial  stresses  surfaces.  and  unless  other  the  were  mm  encapsulant  higher  outside  at  stress the  valleys  near  the  distribution  crystal  in  the  W  top  and  centre tend  bottom  to  is  and be  ends  W-  near closer  of  the  crystal  for  &  stresses  crystal  the  for  is  considered  is valid  vertical  Lower  there  conditions  than  crystal.  23.  changes  axisymmetric  lengths  with  the  transverse  symmetry  dislocations  crystal  shaped  to  At  Length  conclusion  For  the  sections  lower  radii  valid  angles.  growth  crystal  crystal are  the  given  Crystal  The  angle  of  apparent levels  larger  are  lengths  observed  larger  than  crystal  shorter  stress  concentration  crystals encapsulant thickness  there  is  surface. and  than  a  at  both  encapsulant  the  encapsulant  stress  temperature  of  the  below  These  ends  are  the  thickness.  thickness,  the. shoulder.  For  concentration  above  valid  gradients  there longer the  for  any  encapsulant  in  the  environment  297  surrounding be  the  the c r y s t a l .  case  profiles  for  in  the  It should  short  be n o t e d  crystals  encapsulant  as  and  that  this  linear  described  may n o t  temperature  in  paragraph  12  above .  25.  The  stress  maximum The  stress  due  The  this  the  largest  stress  a  common  The  level.  longer  stress  above  In  at  and  the  This thermal  encapsulant  the  the mid-length  a  thickness.  crystals.  thicknesses  conditions  during  reaching  encapsulant position  contribute  to  in  give  growth.  to the  MRSS  longitudinal  cannot  be a s s o c i a t e d  (010) planes  the  most  modes a r e I I I , IV a n d V.  stress  following  a)  level  corresponding  stress  for  with  strain  length  the encapsulant  discontinuity  coincides  quasi-plane  mode  maximum  thermal  crystal  The s l i p  crystal  twice  decreases  the  with  27.  the  When  with  f o r a l l encapsulant  studied. to  surface.  length  slightly  is valid  gradients  26.  increases  for a crystal  result  is  level  There  symmetry  patterns  in  transversal  (001) p l a n e s  shows t h e  :  i s always  eight-fold  symmetry  a t t h e edge  of the  sect ions.  b)  The f o u r - f o l d liquid  symmetry  interface.  i s stronger  It also  appears  closer  to the  a t the seed  end.  solid/  298  c)  d)  e)  28.  Two-fold  symmetry  results  four-fold  symmetry.  It  the  minima  in  Far  from  the  edge  of  ends,  For  crystal  lengths  larger  far  from  the  <100>  the  dislocation growth  The  interface,  directions  dislocation  lower  The  than  far  the  four-fold  by  is  two  than  that  at  the  elongation  except  eight-fold  of  in  and  stress  levels  the  <110>  with  at  the  symmetry.  radii  predicted  shows  an  of  directions.  minimum  rather  the  degeneration  axisymmetry,  than  density from  is  there  distribution  conditions  <110>  there where  a  manifested  and  section  normal  b)  <110>  the  The  a)  the  is  from  sections are  in  directions.  the  model  for  ends  are  :  the  seed  and  tail  ends.  symmetry  always  forms  close  to  the  inter-  face.  c)  There from and  is the  40.0  Growth  a  correlation cone  of  between  stress  two  symmetry  crystals  with  with  distance  radius  27.5  mm  mm.  Velocity  _3 29.  For  growth  ature  velocities  field  velocity velocities  is  on  the  larger  at  of  10  steady stress gradients  cm/sec state,  field and  and  and is  lower,  the  effect  negligible.  stresses  are  the of For  obtained.  tempergrowth larger  299  Effect  30.  of  Increasing little of  For  and  the  of  the  the  gradient  The  radial  not  significantly  27.5  mm  to  accounted  The  higher  radial  creates  in  the  40.0  radius  the  encapsulants  radius  increasing  on  relative  and/or  is  effect  depends  the  The  stress  increment.  low  the  thermal  radius  is  2  increment.  level the  temperature stresses  with  The  a  the  with  change  a  between  the  the  crystal  stress  the  crystal  level  change  level  in A  crystal radius  do  from  cannot  be  gradients.  larger  field  larger  in  in  thermal  gradients. in  surrounding  gradients  the  strain  decreasing  linear.  stress  in  and  relation  temperature  changes  of  thickness  is nearly  mm.  stress  larger  of  environment  change  by  mm)  The  thin  relative  relative  level.  axial  incompatibility from  effect  reduction  for  the  (40-50  the  For  (100°C/cm)  encapsulant  and  levels  There  Conditions  stress  and  stress  to  levels.  distribution.  environment.  similar  than  stress  stress  gradients  Thermal  the  on  the  relative  gradient  decreases  in  is  larger  Increasing thermal  radius  encapsulants  times  Effect  increases relative  normal  thick  3  the  the  increment  gradients  33 .  on  conditions  mm)  level  32 .  radius  increasing  (21  to  the  change  thermal  31 .  Radius  radius a  given  crystal.  is  due  cylinder radial  to  an  arising gradient  300 34.  To  obtain  stresses in  are lower  t h e 2 7 . 5 mm  This  21  by  a  that  Pressure  and  experimentally pressure  crystal likely  modify  and  transfer  less  during  the  gradient  than  normal  accounted  in  a  of  the  the  f o r by  transfer gas.  effects  coefficients  in  10°C/cra crystal.  growth  be  thickness of  gas  thermal  the  between  corresponding  will  change  gradients by  between  of  surrounding  the  observed  coefficient  i s indicated  pressure gas.  The  which  can  across  the  the  encapsulant  larger surface  gas p r e s s u r e .  significantly  region  Curvature  modifies  approximately  The m o d i f i c a t i o n s c o n s i s t  A stress  density  The e f f e c t  the  I n t e r f a c e Shape-Convex  interface  interface.  dislocation  i n the gas. This  gas f o r h i g h e r  ution  to  efficiency  A convex  be  on  of the heat  related  Solid/Liquid  a)  be  f o r encapsulant  and t h e s u r r o u n d i n g  encapsulant heat  of pressure  cannot  significantly  36.  to 8  an a v e r a g e  f o r t h e 4 0 . 0 mm  gradient  of 6  which  Nature  dependence  transfer  and  should  a n d 7°C/cm  the  factor  The l a r g e e f f e c t  is  environment  stress,  in  mm.  Gas  the  the y i e l d  crystal  requires  reduced  (80 * o f t h e c r y s t a l )  than  the surrounding  for  35.  l a r g e areas  the stress  one  radius  distribfrom  of the f o l l o w i n g  concentration at the interface  edge.  the  301 b)  A  U-shaped  c)  The  stress  above  distribution  characteristics  close  do  to  not  the  interface.  change  with  crystal  length.  d)  Highest  stress  interface  for  values all  dislocations  are  crystal  are  obtained lengths.  at  the  This  generated  edge  of  indicates  the that  immediately  after  solidification.  e)  Stress  levels  thickness stress  and  Increasing level  Concave  A  or  is  increasing  the  temperature  not  eliminated  the  by  encapsulant  gradients.  The  increasing  the  thickness.  the  crystal  distribution  interface  distribution thickness  a)  decreasing  by  radius close  to  does  not  the  change  the  stress  the  stress  interface.  Curvature  concave  the  reduced  concentration  encapsulant  f)  are  ' in  from  shape  regions the  substantially adjacent  interface.  The  to  modifies  and  about  modifications  one  radius  consist  of  centre  and  following  Stress edge are  concentrations of  the  observed  the  crystal  largest above  and the  are at  developed  the  about  at  interface.  twice  encapsulant  as  large  surface.  the The as  stress  levels  stress  levels  302  b)  Increasing  the  distribution. to  the  Cooling  The  using  the  has  been  the  boron  1000°C found  for to  the  method  employed  to  38.  results  The  are  :  %  the a  distance and  and  interest  a  the a It  which  the  the  were  was  also  is either  solutions  limit  the  three  and  gradients  the 110 were  The and  The  element  growth.  The  interface  and  interface.  The  mm  respectively. considered.  The  follow.  stress fields close  to  or  were  value.  finite  adjacent  mm  argon  series  region  27.5  assumed  atmosphere  during  were  initial  argon  numerical  from  obtained  temperature.  calculations  radius  during  conduction  parabolic  theoretical  using  Growth  heat  constant  for  the  after  stress  showing  the  of  analytical  a  stress  proportionally  fields  assuming  at  The of  the  distributions  direction.  is  of  length  and  solutions  i n a medium  conclusions  from of  96  stress  1000°C  oxide.  on  temperature  region  or  increases  temperature  axial  calculated  profiles  markedly  the  medium  to  for  a  radius  Temperature main  were  focused  extending  in  change  Temperature  obtained  800°C  converge  analysis  were  boron  fields  The  not  radius.  and  dependent  the  were  in  Ambient  i s immersed  and  stress  crystal  time  crystal  temperatures  to  analysed.  profile  oxide  change  does  level  temperature  Solutions  temperature that  on  radius  stress  Crystal  analytical  equation.  The  relative  effect  cooling  crystal  fields  during  obtained  during  to  the  interface  cooling growth. the  differ In  the  differences  303  a)  Larger  b)  Larger  radial  stress  higher  The  a)  In  after  c)  The  5  to  larger  d)  The  ; the  may  be  not  of  fields  are  :  between  10  four  and  temperature  most  important  The  radial  stresses  gradients  along  case  from  the  as  In  times  20  seconds  is  reached  axial  actual the  cooling  and  obtained  with  combined  with  and  at  the  from  the  stresses.  initial  because  distance gradient  cooling  is,  conditions  temperature  in  the this  the  media  assumed.  the  the  axis  with  stress  stress  These  rapidly  the  on  are  the  largest  in  (001)  symmetries  characteristics  of  dependent  gradients.  the  The  instead  to  axial  larger  the  differs  shape  Larger  distance.  markedly  a)  the  strongly  decreases  constant,  symmetry  of  reached  not  give  level  the  is  The  axial  shorter not  is  and  surface  interface  two  equilibrium  conditions.  stress  of  minutes.  radial  crystal  is  the  level  different  order  starts.  10  stress  the  growth.  stress  cases  initial  of  characteristics  cooling  most  levels  during  largest  after  b)  than  specific  The  gradients.  obtained  are  distribution usual  W  transverse  shape.  planes  during  growth.  :  always  shows  a  U  or  V  304  b)  At  c)  2.75  mm  stronger  and  is  about  half  At  8.25  minima ment the  which  these  four  by  ring  at  symmetry  the  a  two-fold  an  <110>  to  may  edge  is  which  symmetry  elongation  directions  position  two  of  and  closer  case  the  accounted  process.  similarity tail  end  from  to  is  the  a  two  displacethe  seed  a  edge  of  depends  on  between showing  <100>  symmetries support  the  or  The  axisymmetric  <110>  directions.  two-fold present  at  circular  than  observed  LEC  close  of  the  to  disloc-  tail.  In  crystals  during  the  may  cooling  experimentally  in  conclusion.  distributions  two-fold  early  dislocation  from  in  generated  this  an  symmetry.  giving  far  observed  symmetry  symmetry  cooling  larger  EL2  nearly  symmetries  c o n s i d e r a b l e number  during  the  differnt or  nearly  stresses  end  symmetry  eight-fold  densities the  The  symmetry  a  times  two-fold the  i n the  The  that  larger  The  from  evolves  f o r by  time.  the  dependent.  cooling.  four  mm  eight-fold  generated to  8.25  four-fold  suggest be  and  either  a  during end  than  i s time  from  results  densities  the  minima  tail  ations  The  and  minima  eight-fold  interface  two-fold,  symmetry  the  41.  <110>  are  wafers  larger  formed  observed  stage  be  is  the  area.  the  conditions  evolves  this  a  wafer  cooling  The  The  in  distances larger  with  40.  interface  wafer.  For  e)  the  from  i n the  of  the  appears  mm  observed  d)  from  symmetry  in and  wafer  from  sometimes  305  minima  in  the  <100>  directions 80  proposed  by  mechanism climb  as  al.  mechanism  81 '  in  the  for  Appendix  the I,  EL 2  formation.  involves  This  dislocation  cooling.  of  the  mathematical  during  et  described  during  Summary  A  Holmes  supports  growth  Conclusions  model  for  and  cooling  results  describe  temperature of  LEC  GaAs  and  stress  has  been  calculations developed  and  distribution in  LEC  validated.  The  model  GaAs  with  thermal  The  mechanism  stresses  effect  variables  The  a  on is  local  and  presented  show  that  be  temperature  gradients,  8.0  cm  crystals.  established.  large  therefore  of  in  which  only  cooling  the  different  the  crystal  growth  discussed.  stress  importance  formation  d i s t r i b u t i o n of  and  low  The  dislocation  shear  diameter  dislocation  involved.  dislocation  resolved  stresses  for  are  calculations  the  areas  below  free  of  between  on  the  in  critical  resolved  dislocations. 7  and  10°C/cm  dislocation  can  This for  formation  have shear  requires 5.5  has  and  been  306  REFERENCES  1.  L. H o l l a n , J . P . H a l l a i s and J . C . B r i c e ; Current Topics i n M a t e r i a l S c i e n c e . 5.E. K a l d i s E d . N o r t h H o l l a n d . Amsterdam, 1980, pp.1-218.  2.  D.N. McQuiddy, J.W. Wassel, J.B. 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Cook, "Concepts and A p p l i c a t i o n o f F i n i t e A n a l y s i s " , 2 n d e d . J . W i l e y a n d S o n s , N.Y., 1981.  211.  V. S w a m i n a t h a n 1975, pp.482.  212.  H.A.  213.  Handbook and I.A.  214.  R.H. G a l l a g h e r , J . P a d l o g May 1 9 6 2 , p p . 7 0 0 - 7 0 7 .  215.  E. W i l s o n , A I A A  216.  N.P. D a r i o a n d E n g . , 5, 1 9 7 3 ,  217.  R.W. C l o u g h , a n d Y. R a s h i d , ASME, E M I , 1 9 6 5 , p p . 7 1 - 8 5 .  218.  219.  and  S.M.  Nancarrow, Proc.  Copley,  Phys.  J . Am.  Soc., 45,  1933,  of Mathematical F u n c t i o n s , S t e g u n , D o v e r , N.Y., 1965.  Journal,  and  3(12),  W.A. Bradley, pp.573-583.  I . G r a n t , B. S m i t h , Instrument Co., Commun i c a t i o n .  D.  P.P.  Rumsby Dustat  Ceram.  Ed.  by  M.  Abramowitz  ARS  and Rd. ,  the  Journal,  pp.2269-2274.  I n t . J . f o r Num.  J . of  S o c . , 58,  pp.462.  Bijlaard,  Dec.1965,  Element  Eng.  Methods  Mech.  R.W. Ware, Cambridge,  H.S. C a r s l a w and J.C. J a e g e r , "Conduction S o l i d s " 2nd e d . , O x f o r d P r e s s , O x f o r d , 1958.  in  Div.,  Cambridge Private of  Heat  in  320 APPENDIX  EFFECT  I.1  Effect  The use  as  OF  DISLOCATIONS  o f D i s l o c a t i o n on  semi-insulating substrate  simplifying  for  device  ON  GaAs  AND  DEVICES  Properties  property  the  I  of  GaAs  fabrication  isolation  and  makes  of  i t suitable for  integrated  permitting  circuits,  low  capacitance  49 interconnections implantation on  has  been  undoped level,  voltage examined  no  as  GaAs  at  less  '  .  carefully sources  concentration  choice  rigorous  FET  threshold  than on  20-30  how  Recently, and  of  Moreover,  or  sinks ,or  distribution  The  semi-  performance  i n Cr doped  on  a  and  micro-scale  is affected  microns  from  how  much,  the  the  to  is  the  FET.  has  been  interaction  defects strongly  by  threshold  correlation  point  which  ion  voltage  this  related  of  conditions  homogeneity.  Inhomogeneous  .  t h e FET  consensus 54 55  affected  more  dislocations  with  located  been  is  that  The  dislocation distribution 5 0-53  semi-insulating  has  level  with  .  imposes  especially  is essential.  correlated  i t i s found  substrate  fabrication  properties,  property  dislocations There  the  f o r device  substrate  insulating  through  and  the  of EL2  correlated  FET c h a r a c t e r i s t i c s . In  additions,  a great  amount  of e f f o r t  has been  directed  to  evaluate t h e r o l e o f d i s l o c a t i o n s on s e m i - i n s u 1 a t i n g p r o p e r t i e s . Dislocation d i s t r i b u t i o n s have been directly correlated with i i 4. «- - K <-• 56,57 . ... 58-61 leakage current d i s t r i b u t i o n , resistivity , and c a r r i e r  321  concentration  in  Si  implanted  LEC-growth,  semi-insu1 a t i n g  6 2 64 GaAs  '  .  Leakage  distribution, the  W-shape.  the  sheet  current  while In  and  resistivity  addition,  carrier  sheet  i t  and  is  concentration  resistance carrier  found  have  M-shaped  concentration  that  dislocations  within  about  reports  are  a  75  have affect  micron  radius  64 area In hand, of  Cr-doped  LEC  i t i s found the  behaviour  Cr  follows  EPD  can  of  the  that  W-shaped  dislocations  GaAs  distribution,  affect  donor  a M-shaped  the  Cr  conflicting.  On  distribution  inverse  suggesting  concentration  i m p u r i t i e s ^ . On  the  other  one  that  and  hand,  the  annealing  no  variation 6 6  of  Cr  the  concentration resistivity  Moreover  wafer  to  does  I t was  follows  concentration added  variations  resistivity  concentrations. a  associated  follow  not  also  a  with  a  the  reported  with  that  Cr  Cr  found  the  mechanisms  except  while 6 7  distribution  that  determine  at  is  out  high across  the  Cr  the  and  distribution.  to the m e l t to o b t a i n s e m i - i n s u 1 a t i n g GaAs. At p r e s e n t , e x t e n s i v e s t u d i e s are b e i n g c a r r i e d  understand  ,  concentration  distribution,  W-shaped  is  W-shaped  correlate  U-shaped  follows  dislocation  Cr++  usually  in  order  semi-insu1 a t i n g  6 8 properties materials dopants donors  of  GaAs  depends present.  is  .  The  on  a  In  the  compensated  production  critical  by  compensation  four-level an  of  model  excess  of  model  an  deep  an  high among  resistivity the  excess acceptors  different of  shallow  over  deep  6 9 donors  .  donors  is  In  the  three-level  compensated  by  the  deep  e x c e s s of a c c e p t o r over 7 0 donors . The shallow donors  322 are of  believed Te.  ubicated  deep EL2  undoped  to  the  EL2  four the  and  Si  are of  i s due  family.  or  due  Cr  and  are  level  important  intentional  to  C,  donors  and  valence  the  doping  or  other  acceptors  are  respectively.  deep  donor  levels.  mechanism  mechanism  Mn  band  mid-gap  compensation  three  the  mainly  and  to  Both  to  shallow  conduction  level  most  S  levels  acceptor  GaAs  The  to  energy  or  the  due  acceptors  The  close  The  GaAs  be  Shallow  impurities.  called  to  i s the  For  applies  so  Cr-doped while  for  applies.  commercially  available  SI  substrate  is  71 Cr-doped.  However,  detrimental  the  large  effects  in  diffusion  the  IC's  coefficient technology.  of  Cr  has  Attempts  to  72 substitute unlikely  Cr  f o r the  because  one  order  Vanadium  of  magnitude  plays  no  slower  Vanadium  direct  role  is  in  the  lead  to  7 3 compensation  process  concentrate  the  understanding summary  of  efforts  the  the  interaction  of  was  the  Martin  Alternatively, distribution showed EL2  the  origin  presence  the  of  et a  is  EL2  due  stress.  role  on  EL2  with  al  have  This  two  GaAs  EL2.  facts  and  In  given  consequently  this  review  below  associated not  due that  correlation the  W-shaped a  is  shown  strong  to  SI  of  was  i t was  ;  These  a  because  on  short of  the  dislocations.  level  that 7 6  .  undoped  and  EL2  found  typical is  on  nat ure  demonstrated  addition  GaAs  findings  Originally it  in  EL2  was  this  i t i s not with  concentration  distribution.  complex  to  It  lattice  supported  73 oxygen until 74 76 element ' In  with  the  the  to  Cr.  dislocation  along  the  i s proposed  defect by  related  growing finding  wafer  that  the  in  the  that  the  323  EL2  level  that  i s created  antisite  i n GaAs  defects  during  form  mechanical  during  77  deformation  plastic  and  deformation  by  78 dislocation As(Ga)  climb  .  antisite  addition,  the  or  a  then  complex  s t o i c h i ome t r y o f  correlate  with  transition  from  critical  I t was  As  the SI  SI  f o r As  is  that  containing the melt  properties  of to  of  0.475  atom  also  observed  during  The  proposed  reaction  EL 2  i s the 79  antisite  In  GaAs  the  melts  the  the  i n LEC  rich  concentration  transition  proposed  i s shown  material  to  with  a  p-type  m a t e r i a l at the 80—82 fraction . This  annealing  of  SI  undoped  Ga  83 rich  melts  through d i s l o c a t i o n V(Ga) + A s ( A s )  The  most  striking 84  EPD  distribution distribution  from  the  tail  minima  are  two-fold (110)  (b).  been  EL2  level  the  proposal  two  appears  by  antisite  level  distribution  the a n t i s i t e  and  EL2  1.1 from  typical at  (a.b) where  with  the seed  end  four-fold the  the  symmetry  seed  end. four  <110>  at  seed  with  second,  a mirror  plane  the  coincident  d i s l o c a t i o n s at  . Nevertheless,  proportional  is  local end  a  with  a  the  seed  end  the fact  that  the  t o d i s l o c a t i o n s has  concentration  of  It  : first  and  EL2  (a) and  f o l l o w i n g aspects  Weinberg  the  density  f o r wafers  symmetry of 33  i s not d i r e c t l y that  i n Figure  The  i n the  Two-fold  reported  o f EL2  perfectly  the  observed  has  present  i s shown  to note  plane.  dislocation  i s shown  end  symmetry  the  85  correlates  interesting  to  is : > As(Ga) + V(As)  correlation  '  level  dislocation  climb  leading  l e d to  i s proportional  to the 86 climb . At  t o t h e amount o f d i s l o c a t i o n 90 91 94 95 has been f o u n d ' ' ' and shown  to  be  324  Figure  1.1  EL2 c o n t o u r p l o t t a k e n a t t h e s e e d o f a ( 1 0 0 ) 3 - i n - d i a m LEC G a A s c r y s t a l . The s o l i d d o t s i n d i c a t e positions where absorption measurements were made, and the contour lines w e r e d r a w n t o be c o n s i s t e n t w i t h t h e measurements. The number assigned to each cogtoug i n d i c a t e d t h e E L 2 c o n c e n t r a t o r ) ^ i n u n i t s o f 10 cm ( i . e . , 2.05 m e a n s 2.05 X 10 cm ) . The c r o s s - h a t c h e d areas i n d i c a t e l o c a l maxima.  325  Figure  1.1(a)  EL2 c o n t o u r plot at t h e t a i l end of the crystal d e p i c t e d i n F i g u r e 1 ( a ) . T h i s EL 2 d i s t r i b u t i o n has a m i r r o r p l a n e c o i n c i d e n t w i t h a (110) plane.  326  stable of  at high  t h e EL2 l e v e l The  also  believed  t o be  the  origin  ft 9 .  enhanced  been  97 93 ' and  temperatures  concentration  o f EL2  level  a t t r i b u t e d to the aggregation  a t d i s l o c a t i o n s has  o f EL2 c e n t e r s  similar  to  87 a  Cottrell  atmosphere  .  enhancement  i s , i t has been  fundamental  role.  EL2  This  insulating  the  demonstrated  there  i s based  mechanism  that  i s not  mechanism  on  complete  for  [EL2]  d i s l o c a t i o n s play  the d i r e c t 88 at s i n g l e d i s l o c a t i o n s  concentration However,  Whatever  observation  understanding  of  a  of the  the  semi-  i n GaAs. 9 6 97  1) T h e r e  i s more  2) A m i d g a p capture 3) T h e  than  level  one m i d g a p  r e l a t e d to  oxygen  c r o s s - s e c t i on h a s b e e n  characterization  inconsistent the  trap  mid gap  1  0  0  ,  1  0  level  1  ,  with  techniques  been  "family")  larger  identified of  specifically,  have  ( a n EL2  9  8  '".  EL2  have  different  obtained  in  electron  been  response heat  of  treated  10 2 horizontal 4) T h e r e  B r i d g e m a n grown  i s a close  sheet  correspondence  and r e s i s t a n c e  (opposed  to  a n d LEC g r o w n  what  between  and an i n c r e a s e is  GaAs  expected)  the increase i n  i n EL2  in  concentration  dislocation  free  . 10 3 GaAs  The e f f e c t s  EL2 1  0  8  ,  1  understanding  level 1  0  .  has been  This  associated  reinforces  of the o r i g i n  with  the  and p r o p e r t i e s  undesired  need  for  of t h i s  backgating a  level.  complete  327 The  presence  of  carbon  in  GaAs  has  been  shown  to  affect  104 directly  the  threshold  concentration  in  LEC  voltage  GaAs  of  depends  implanted  FET  (carbon  the  content  of  the  cell  walls  of  on  water  10 5 encapsulant the  ).  cellular  that  arrays  oxygen,  segregated I .2  Carbon of  the  and  silicon  dislocation  D i s l o c a t i o n s and  Dislocations  in  GaAs  level  I.2(a-c).  by  In  dislocation the  (c)  the  the  below  of  the  migration  of  V(Ga)  In  this  vacancies  native  in  are  midgap  of  atomic  with  GaAs  a  shift  of  doped  Note effect  i n which  The  As  to  the  V(As)  The  are  on that  shift  is  1.2(b)  and  shown of  as  a  dopant  is  n-type  doping  i s accounted  for  by  Ga  may  be  also  be  the  vacancy  vacancies  which  can  be As  can  created  upon  site,  As(Ga)  As(Ga)  (V(Ga) in  Figure  Note  Figure  that  neighbouring +  GaAs.  fraction. In  in  composition  concentration  regime. This  melt  density  The  antisite  levels  shown  undoped  illustrated the  generation,  >  As  is  measurements.  mechanism  + As(As)  been  .  lightly  density.  vacancy  has  in  correlated  This  for  defect.  the  4 3  dislocation  Ga  reaction  effect  the  dislocation the  the  hardening  condensation  important  .  and  dislocation  critical  '  concentration.  impurity  impurities  been  0.500  of  energy  carrier  the  vacancy  above  i t  12 7  shown  uncertainty  the  reduces  is  occurs  Fermi  function  1.2(a)  density  minimum  within  a  Figure  the  Level  have  Lagowski  at  Also  clusters  Fermi  12 6 Fermi  segregated 106  dislocations  chromium  to  i s found  the  is  also  lower  created.  h a l f and  V(As)  Both in  Arsenic fraction m melt,X 0.495 0.500 0.505 0.510  328  fl5  I0  6  0 p-typ€ • n-type  I0 611  _|_  2  613 615 617 619 621 623 Arsenic source temperature,T (°C) As  Fig.  1.2  a  Dislocation density vs a r s e n i c source temperature ( m e l t s t o i c h i o m e t r y ) f o r l i g h t l y d o p e d n - t y p e and pt y p e GaAs  I0  17  I0  16  I0  16  I0 I0« 17  Carrier concentration (cm-3) Fig.  1.2  b,c Dislocation density v s 300-K free-carrier concentration f o r c r y s t a l s grown from optimum melt stoichiometry and under optimal thermal stresses. U p p e r p o i t i o n shows t h e c o r r e s p o n d i n g values of the F e r m i e n e r g y a t 1100-K.  329 the  upper  Fermi  half  level.  migration  by  the  n-  the  The  of  dislocation  of  amount  the  The  p-type  states.  In  place  between  charged  The  rate  lightly  concentration.  than would to  the  however,  p-type  to  the  material. i s not  the shift  with  the  opposite The  reaction  the the  on  the  determine  the  to  coalesce  Fermi  level  above  into caused  f r a c t i o n of  the  the  +  +  neutral  reaction  takes  third  3e-  power  density  that  the  V(Ga)  involving  behaviour  exact  would  + V(As)  dislocation  i t i s assumed  known.  of  depend  states  As(Ga)  decreases  will  ability  material  >  since  states  increases  doped  Since  V(As)  lead  p-type  to  and  (occupied)  reaction  occupancies  neutral  downward  + As(As)  + +  of  transition  of  transition  V(Ga)  V(Ga)  Their  vacancies  loops. to  band).  with  mechanism  for  electron  increases  upon  migrates  rather  this  the  of  type  of  transition  vacancy  defect from  n-  condensation,  330  APPENDIX I I  Integration  by P a r t s  of the Element  Equation  f o r the Temperature  Field  A useful  formula  f o r the integration  by p a r t s  c a n be  written  as [ 2 1 0 ]  N P,  dD  (e)  N, £ P d D  ,(e)  +  ( e )  N P n ^ d S  (e)  D  (  e  (II.1)  )  (e)  and '  N 1  ( P), dD P  (e)  Where  N  indicates normal  n  and  P  N.  ( e )  D  ( e )  +  N P n dS * * (e)  functions  n^  and  to the elemental  (II.2)  C  (e)  are given  derivative,  P dD  n  p  surface  o f £ and p , are director dS  v  and  the  cosines  comma  of the  ' i n t h e £ and p d i r e c t i o n s  respectively. The  integral  (e) Using  3p  1  2  1 D  P  t h e above  N . (e)  to integrate  1  P  36  3 6  3p  3C  formulae  3  i s:  2V  for integration  dD  D  _33 3p  dD  n  dS  (e)  (e)  (e)  by p a r t s ,  3N,  (e)  3p  .(e)  33  2  33 P  3p  by p a r t s  3p  0  (II . 3 )  we g e t  33  dD  ( e  >  +  3p ( I I .4)  331  and  3 (e)  1  33  3?  dD  3£  33 (e)  3N  (II. 3),  33  i  3p  (e)  in  3N  3p  n^dS  33 -  3?  2V  (e)  the  boundary  3N.  3p  ,(e)  +  r.  .  33 3p  (e)  dD<  3C  n^  3?  condition  (7.3)  33  3N  N.h(z) l  that  e )  33  3p  +  (II.5)  33 N 1  33 1  35  +  3C  3  ( 6 )  ( 6 )  (e)  i t i s obtained  3C  3C  , . dD  H  1  A  N .  Using  3C  (e)  N  Substituting  33  3N.  (e)  we  2V  finally  ( 6 )  =  (e)  get  that  33 N  r  dD  3?  1  dS  dS  +  0  N.h (e )  ( e  >  z  +  3  z  dS  (e)  332  APPENDIX I I I  Evaluation  a)  of  the Matrix  Evaluation  of  Elements  f o r the  Temperature  Field  K, ij  Making  dD  the  =  ( e )  2  following  pd  TT  substitutions  we  3L.  c. 1  35  2 A  2 A  get  = T  (7.10)  pd £  3L. i_ 3p  i n Equations  i j  2 TT ~ (2A)  (b.b. +  2  Substituting  1  J  c.c.) 1  the  natural  I p,L. k=l c o o r d i n a t e s we  Trp( >  and  using  A  i J  where  (e)  2  (  I P, ) i =l  tables  2 TT j  L .pdpd 5 ;  (2 A) J  f o r the  integrals  k  get  :  e  (b.b. i j  2Vb  j  p =  k  p dpdC +  +  /3  c.c.) i j  +  *  V  h  l 1  6  [  3 p^  (e) ' +  p. J  ]  of  333 b)  Evaluation  of  K " i j  These of  the  elements  external  surfaces  with  of  to  to  the  Vertical  surface  Consider  an  the  the  respect  (perpendicular  i)  evaluated  surface  (parallel  (inclined  are  to  for  three  different  peripherial  crystal the  elements  axis)  axis)  and  ;  orientation  : i)  i i ) tilted  iii)  vertical surfaces  horizontal  surfaces  axis).  element  like  the  one  i n the  figure  k  z  r  The surface  only i-j .  variable  K  =  H . . i J  where two  £  c o n t r i b u t i o n to The is  r h(z)  nodes  the  dS  (e)  =  the  distance  natural  integral  surface  2TTpd£  L . L . dS i J  Q  1. . i s i3  elemental  the  (e)  with  =  area p  =  (7.10c) in  terms  constant.  r h(z) Q  2iTp 0  between  coordinates  nodes can  be  i  comes  and  written  of  from  the  the  local  Therefore  L . L . d £ i J  j . Between as  these  334 Substituting  i *  and  integrating  we  get  j  2TTh(z)  r p Q  i j  i  = J  2TT  h(z)  1. LI  r p Q  ij  ii)  Tilted  For surface  an area  Surface  element  with  c a n be w r i t t e n dS  where  £  tilted  i s the l o c a l  (e ) i e ;  external  surface  as 1  =  2TTp.dc-  coordinate  as shown  i n the  Then  2 TT r h ( z ) Q  ij  the  ij L.L.nd; i J ^  figure  elemental  335 p  can  be  written i £ as  coordinate  as  a  function  Pi  +  p^  of  < Pj "  ,  P^  p^  and  the  local  1 . . i J  p ,  Substituting integrating  we  and  for  the  K  27Th(z) r  ]  H  i  p  1. . ~ 12 i  Horizontal For  elemental  The  an  +  element area  coordinates  with c a n be  +  horizontal written  are also  L. l  p .  +  i  j  p. J  =  external  as  written  as  1 i J  and  p  M  p.  j  p  i  +  3  p  j  Surface  surface  natural  and  U  p .  iii)  expression  get  3  [  above  surface  the  336  l J  —c by p  Approximating p  K  =  Pj ) /2  2 TT r h ( z ) p  =  H  ( Pj +  rl  (S)  Q  ij  LJLJ  dp  i j Integrating, i  the elements are  = j  K.  (S)  2TT r h ( z ) p Q  ij  i *  j  2  TT  r h(z) p  (  S  )  — ^ 6  b,  f o r dS^ ^,  i j  c)  Evaluation  Making  of Elements K  t h e same  substitutions  as i n  e  L. a n d L . i  and p i n (7.10d)  i)  Vertical  f o r the three  Boundary  Element  TT  ii)  Tilted  Boundary  {K > A  =  TT  c a s e s and i n t e r a t i n g  h(z) r  Q  6 (z)P l a  u  ^  1  Element  h(z) r  Q  6 (z) (P a  A  +  2 p )  J L L  we g e t  3  337  iii)  Horizontal  Boundary  <V  Element  " * <«> 0 a h  r  6  ( z )  P l  ij 1 I  338 APPENDIX IV  Evaluation  of S t i f f n e s s  Substituting multiplying e  Matrix  the  the four  Elements  expressions  elements  f o rLinear  for  Elements  [B ]  and  1  of the s t i f f n e s s  [C']  matrix  of  element  are :  1 k  =  [(1 - v ) b b . +V2A (2Ar  1  L L  1 k  1  ?  =  (2A)  1 £  [vb Z  b 1  + v2A  k  0 1  ,  1 "  77772  2  J  1  force  vector  c c ] pdpdC  1 - 2 v  -  +  c.b  2  i i  1  ] pdpdc:  J  1 - 2 V  L  + p  I d - V) c . C j  +  c.b.] pdpd£ 2  —  b. 2  components a r e  +  1 - 2 V  J  1  1 - 2V  I 2A }  The  J  [ v b .c. + v 2 A (2A)  p  J  -  1  p  A  p  C  =  2 1  22  —  J  1  k  L. r— +  c .L . r  b L —  L  J  + (1 - V ) ( 2 A )  b L - - + v 2A  b j  ] pdpd£  and  3 3 9  Also  the i n i t i a l  force vector i s  b.  +  1  L . l  A  2  •  2A  Using  t a b l e s f o r the i n t e g r a t i o n of area  1  k  = 1  -v) b b  [(1  1  1  -  coordinates  c c ] 2  1  + 4  J  V  (b . + b . )  1  k  = X  [vb c  d  1  -  +  21 d  =  l  [  V  b  i  C 3  i  -  C  22  =  n i - v ) c  i J  2  i  C  4  j  - v) A i . 1  c  A  -j  6  J  V b  i  ]  +  C  -  i  6  4A  1  1 k  +  2V  +  1  (1  v  J  J  2  3  1 k  i  b  A  L  +  P  2V c  1  2V  b.b J  +  2  we g e t  p  2V  +  J  +  p dpdC  1  j  4  A  L  dPdC  340 APPENDIX  Quadratic  The matrix  Element  Calculations  stiffness  with  three  V  matrix  f o r the  different  quadratic  element  is a  12  regions  I  [k [k  1  [k']  3  1  [k-  [k']  4  1  [k  1  [k']  5  1  [k  1  [k']  6  1  [k-  ^ 22 j 32  ]  4 2  ^52 j 62  1  ,13  [k'] [k  1  2  Symmetri c  3  ^33  [k' l~k' Ck'  [k j 53  [k  ^63  [k  ,44  ]" ,64  [k«]  The well  [k  as  the  force  the  vector matrix  5  [k  [k']  6  5  [k'  2 X 2  matrices relating  f o r any {F  element  i s a 12  nodes X  1 (e ) } o  11 w.  (d>  (e)  j 56  5  1  j 66  11  •jqp a r e  displacement  tk'  [k']  111 the  4 5  12  w.  61 62  q and  1 matrix  as  x  12  341  The  Interpolation  area  functions  coordinates  element  are  4  L  i  (2L L  i  -  i  i  1)  1,2,3  i + 1  l  +  =  form  for  q <  from  the  (e) of the [ B ' ] v  matrix  f o r node  and  has module  i.e.  The  built  as  L. 'i + 3  f o r the  1 = 3  q depends  three  ;  1 + 1 = 1  on t h e v a l u e  of  3  L  q  <  2 L  (4 L  4 L  q  q  -  *> 1)  2A  and  for  q > 3  V i +l 2 A L  +  L  i i 1 b  +  i i l L  +  [B ] 1  (e)  3+i  2 A  c .L. , + L. c. . l l+l i l+l c.L. „ + L. c. , l l+l l i+1  again  i + 1  has module  3  b .L . , + L .b. . l l+l i l+l  342  Using  these  performing stiffness  a  )  the  [B ]  integrations  modal  matrix  R e g i o n I , nodes Case  1  matrices  1  in  we  Equation  get  the  elements in  11  f o r the  regions.  q = p  - v)  b b q  +  Z p  p  +  _  ( 1  ,qp 12  ( v b c q p  +  .,qp '21  [ v c b q P  +  P  . b  c Q  2v  q  +  - 1) L E  1 1 L  . ) (  +  Z p  (b  P  2 2 L  +  E p K  ( 2L p_ P  -  d  3 3  V c  n  p 15 K  vc.  10(2A)  15  Z p M  b b ] q P  n  +  1)  L  + 2p  n  + b  15  + 2p _q . =*—) + 10(2A)  P  2v p  +  10(2A)  (2L a  c b ] P q  v 3-  q  L | j i/ _ a  v )  2v  [ ( 1 - v) c c  + 2p  c c ]  P  22  values  and  1 to 3  2  =  following  the three  l-2v [(1  (7.25c) o p e r a t i n g  2p  z_q_  10(2A)  L  l  d  L  2  )  343  Case  2  q * p 1 p  1-2V  V k  >  l l  [ (1 - v ) b b q p  qp  +  c  + p + p _g ( 30(2A)  n  c ](• p q  v (b  + b q  30  ) + p  L +  (1 -V)  |J| /  (2 L  9  P  12  c q  L  [ vc b q P  ..qp '22  [(1  P  2 2 L  c b ] ( 2  . ,qp 21  +  q  I p c b P  3 3 L  q  V c ~) 30  + p + p ^ — —) 30(2A)  ]  I p b  b Q  q  v c  9  Q  v ) c c + q p  I I , Nodes  dL.dL. 1 2  30(2A)  2V  b) R e g i o n  P  E  P  2v  +  +  - 1 )  Z p + p + p  +  P  (2L E  E  1 1  1 - 2 V [vb  - 1) L  9  + p  + p -3-) 30(2A)  (—-S  ]  30  ^  P  = j + 3  q, p > 3 p = i + 3  Case  k  l l 3  1  1  +  1  '  3 + J  1  =  i = j  —  1  15  l  2 A  i  1 - 2  V  +  2 v ( b  i  +  b  i l +  b  j  +  b  J + l>  I'  +  + 16 2  +  2A  (1 - v ) I .  +  344  ,3+i, 12  3+j  ,3+i, '21  3+j  .,3 + i , 22  3+ j  2  I  V  1 - 2V  14 +  2  < j  V  C  2A  15  I  V  15  41  +  +  j + 1>  +  t  c  2A  )  1 + 1  2A  -  ) H  V  1  +  I  15  1  2A  Where  ;  =  2b.b. ( Z p  +  b  i j + l b  ( Z  J|  1  1  + 2p.  n  Pn  +  2  14  l  +  I' 41  =  b  i  C  j  (  b  i  C  j +l  Z  p  n  (  +  Z  p  ' p  2  n  )  Pi+1  +  1  Z  + 1  J  +  + 2b  i  Pj>  +  dL  1  +  1  +  l d  b  L  b  j  +  (Ep  1  i + l j b  (  Z  +  n  P n  i  b  j +1  (  Z  p  n  )  Pi  +  P j + 1>  +  n n  i + l  +  p  )  +  i +l  2  +  b  p  i+l j+l C  j>  +  ( Z  i +l  b  c  Pn  j  K  C  P i  2  +  2 p  CPn  +  v  P i+l  +  Pj>  +  c  i +l  b  +  K  j  (  J  P  i>  +  2 c . b , (Zp + 2 p , .) + 2 c . , b . , ( E p I j * *n » i +1 l + l j+1 n  +  2  L  p  I  14 2A  1 - 2v (1  41  +  1 - 2v  (c  2V  C  I  + n  i  p  +  p  j +l  )  2p.) i  w  Pi  +  P j i> +  Case  2  i * j  * i+ 1  j  , 3 + i , 3+j  Ij  11  + 1 = i  + v(bj +  2b  2 A  15  2b  i+1  V.  1  I'  1 - 2v  + 16  (1 - v ) I.  2A  ,3+i,  VI  3+j  12  3+ i ,  15  + V(2Cj  +  c  )  41  +  2 A  Vi  3+j  2A  41  2v  + v ( c . + 2 c . ,) + l l+ 1'  14  v  21  3+1,  2v  14  15  2A  2A  2v  3+j  - V ) I£ +  (1  22  1 5 ( 2 A)  Where I"  1  b  i  b  j  ( Zp.  p  + b . b . , ( Zp 1 j+1 n v  K  i +l  K  p  j  +  l  }  b  i+l j+l  + p. . + p.) + i+1  K  j  b  (£  P.  2 b . , b .K ( 1+1 j n  p  Zp K  i  +  j  2p.)  )  346  l  U  =  b  i  =  C  Case  ,3+i, 11  Z  p  b  3  Pi  +  ( Z  Pn  (  1  +  P  +  K  Z  p  n  P j + 1>  +  + p.  (Zp  i j+1  • 3+j  n  +l  c.b. i j  +  k  (  Vj  +  I" 41  i  C  1  i +  +  b  Pj>  +  i  l i C  +  +  2  b  l  +  i + l  (  i  C  (  . + p. ,) + c. „ b .  i +1  P  +  K  1  l  +  j +1  i + l j+1  Pj)  +  +  j * i * j+1  2  c  i + l  Z  n  Z  Pn  Pi  +  +  Pj  )  +  Pi)  + p. + p.) + i  w  K  EPn  +  2  . (Zp n  j  b  p  +  2  j  H  Pi>  i + 1 = j  1 - V  2  +v(2b.  =  15  2A  1  + b.  1  1 - 2v  I  4  +  +  2  1  1  6  (1  + b.  +  1  _  v  J  + 2b, J  +  ,)  1  j  }  2A  "i t  3  l  +  i ,  3+j  2  k  2  V l  .  M  +  15  2  £  3+j  2  V l  = 15  l  V  (  C  J  2A  II  u,3+i. k  ni  14  +  2  C  J  22 "  . )  ~ V  1  2  2  II  +v(2c  + c  ,  1  2  v  )+ 2  1 ( 1  15(2A)  2A  i  41 2A  "  *41  +  J  2 k  , j +l  "  V  )  I  4"  +  2v J  2  4'  i  *14 2A  +  347  Where I"' 1  v  +  I" 1 14  + P.  b b (Zp y i j n  =  2  b  i  j +l  b  (  Z  n  p  +  -  C  i  j  b  (  Z  p  n  +  p  i+l  V  Case  +  k  •a l  i+3  b  (  Z  n  p  +  n  y  i  p  +  p  i +l  j + l  )  +  C  j  ^  i + l j b  K  j'  Nodes  l +1  q< 3  j+ l  p  (  +  n  Z  p  n  +  j  K  j'  )  + p.) +  +p. "  i  K  j  p. + p. ,) i  K  i  p  w  i  M  c . , (Zp j +l ' n  l+l  j  + p. + p.) +  (Zp  v  + 2 p. ) + c . . b . (Zp  M  I I I , Mixed  1  i +l j  b  j+1  K  + 2 c . b . , (Zp i j +l n  ) Region  b. i +1 j+1  + 2 p. ) + b . , c . (Zp  K  ll  +  * i +1  K  + 2 b . c . . (Zp i j +1 n  l  j>  2 p  ) + b.  +  + p. . + p . ., ) + b.  b . c , (Zp i j n  =  + P  i +1  M  +  " j +1  p  j>  +  + p. + p . ,)  n  K  ;  p = 3+i  q  "  I  j +1  H  ; i =  1,2,3  q = i  1  "  1  v  "  \ 2A  30 1 - 2v  *1  +  V  (  3  b  b  i  +  2  b  i +l  )  +  4  c I  4  + 4(1 - v) I  5  2 A  . *\V 1  2  1  n  -  —  30  V —  2A  , V  1 - 2v +  Q  4  V(2c  i + 1 1+ 1  - c.) + 1  2  c I ' - 3 — ^ 2 A  348  •q.i+3 k ;  =  o  c 30  2 1  'q,i+3 H  K  2  ' , I , + 3Vc  1  2A  q  1  1 =  ( 1  2  _  v  )  V  ,  2A  1 - 2v  +  q  —  u  2  ,  c  q  b  +  q  30(2A)  2  - V  1  2  4  ,  i  b q  1  where  I  1  =  1 1  A 4  (3b. . - b . ) Z pw l+l l n  + wp  q  ( 3 c . , - c . ) Z p +p i+1 i n q  =  K  L  .  (2L  K  (4b. + l i b . l i+1'  -  . ,b. i+1 I D  M  - n . ,c. i +1 I  ( 4 c . + 11c. J i i+1  M  - 1) L . L .  »  M  y  L » L  D  L  I  D  L  2  2 Pn n L  Case  •a 3 + i  2  q *i  1  i + 1 *  "  ( 1  V  )  =  b 30  1 1  2A  l - 2 v  q  I . - v ( b . - b.  Q  c  1  1  1  +  + b  1  I " + 4(1 - V ) I  9  2A  k  •q,3+i l  &  1  V  = 30  b  a  I  4" 2A  ) +  Q  K  5  , - V(c  + c  ,  1  ) +  -  2  ^  C  2  q  V 2A  349  K  21  '  3 +1  = — 1-~  v  c  l  " _  — a _ J _  30  l - 2 v v  c  +  q  2A  o  b I " g _ j _ 2A  1 -  .'q.3 + i 22  (1  30(2A)  - V)  c  q  I  4  2  V B  +  1  1  q  i  where  (b. l  - b. ) (4p -Zp ) i+l q n K  ( c , - c. (4 i i + l ' q v  Case  .'q.3 + i 11  -Zp ) *n  v  3  i * q  ;  H  ( p . b . . + p. , b . ) x i+l i +1 I M  w  i + l = q  1 - v b 30  2A  1 - 2v  q  I  1  + v (3b  +2b.  q  l  - b ) + i+1 J  1  c I  4  2A  ' q,3 + i 12  ( p . b . , + p. ii + l i+1 i  K  + 4(1 -  V  ) Ig  1 - 2 b 30  2A  q  r  + v(2c. - c . ,)+ 4 l i+l  V  c  9  I 2A  1  "'  350  2V  q , 3+i 22  (1  - V) c  30(2A)  i" ' + 4  q  q  i  where  •i i  I1  I " 4  d)  -  (3b, i  - b . .) Z p l+l n  <  "  3 c  i  c  Calculation  Substituting multiplication for  i + l  )  Z  P n  Pq  +  of force  ( l i b . + 4 b . ,) - p . b . , 1 l+l i l+l K  ( 1 1 C  i  matrix  +  4 c  i+l>  elements  [B'] and [ C ] i n t o  Pl i+l C  F ' q 0  (7.26c)  1  and i n t e g r a t i o n  -  and p e r f o r m i n g t h e  we g e t :  q < 3  < o> F  for  + p q  K  l  3 P q q  q > 3  +  c.(Zp  + p.  i  < o>  ,) + b . , i+1 l +1  b , x( Z p  n  p.  (Z p  +  "n  l  .)  F  i+3 i  n  .)  " i +1  +  c.  ,  i+l  (Zp  ^  n  K  +  p.)  i  +  2A  351 e) S t r a i n The  Components  strain  displacement  components  as  a  f o r the c a l c u l a t i o n  function  of s t r e s s e s  of  the  a t node  reduced  q i n element  e are  Case  1  q <  1  ~2~E~  p,q  + 4b  u E  -  "  Q  [3b  u ,  - b  red,q  q  q+1  , u  .  red,q+1  ,  -  b  q+2  .  u  ,  red,q +2  , u. + 4b . u„ ,1 q+1 3+q q-1 3+q-l J  , red, q  _  P  3  P  q  e_  ' _ ^' p  =  q  1 2 A  [3c  + 4c  ., w . +4c . w . „ ., 1 q+1 red,3+q q-1 red,3+q-l  =  w  q  r e (  q  i+1  + 4(c  +  4  (  C  - c q  +  l  , w . « - c „ w „ + red,q+1 q+2 red,q +2  n  1 [3(c 2~A~  + b  , J.q  w  u . red.q  red,q + 1  )  + b q  w . red.q  — (c q + 2  u  ., u , „ + b w q+1 red,q+3 q+1  q - l  red,q-l+3  +  b  red,q +2  (c. , u , l + l red,q+1  +b  , „) + red,q+3  H  U  ) -  q - l q-l+3 W  ) ]  q+2  w  red,q +2  ) +  352  Case  £  P3 +1  2  "  q > 3  2 V  [  i  b  ;  "red,!  q = 3+1  +  b  ;  i =  1,2,3.  i +l " r e d , i +1 ~  i +2  b  U  r e d , i +2  +  + 2 ( b . + b . ,) u + i i+1 red,i+3 J  +  r o.i s.,3 + i  2  1  =  -  2  i - l  b  1  ( u  red,1+1+3  J  O  +  U  red,i-l+3  ) ]  [c.w j i + c. , w , . . - c . „ w , , « + i red.i l+l r e d , i +1 i+2 red,i +2  + 2 (c . + c . ) w _ , + l l+l red,i+3 H  + 2c  Y  ^ • o p 4 , i +3  = 2  +  1 ^  b  i-1  v  (u red,3+(i +l )  +w  i l red,3+(i-1)' J  [ c . u , . + b . w , . + c . .. u . . « + l red.i i red,I i+1 red,i +1  i +l  W  r e d , i +1 ~  (  c  i +2 " r e d , i +2  + 2 [ ( c . + c. ) u , . „ + i l+l red,i+3  +  C  i-l  +  b  i - l  ( U  red,(i+l)+3  ( W  red,3+i+l  +  +  W  U  +  b  i +2  W  r e d , i +2  )  ( b . + b. ) w . „ + l l+l red,i+3 H  red,i-l+3  red,3+i-l  ) ]  )  +  ]  J  +  353  APPENDIX  Calculation  The <011>  12 s l i p  glide For  plane,  plane  system  a  coincident  the  of Resolved  are given  the  for instance,  other  three  Stresses  and d i r e c t i o n  crystal with  Shear  VI  combinations  i n Table  grown  in  Z-axis, a r e shown  (111) planes  [001]  three  direction  directions  i n the Figure.  c a n be  {111}  V.I.  the  the  f o r the  vhich  in  the  is  (111)  Similar figures for  drawn  x The stress step  first tensor  consists  one  of  the  other  on  that  stress the  the  plane  tensor  onto of  axis  axis  to  step,  be  for operational a  cartesian  rotating  the  coincident  reasons,  coordinates  cartesian  with  the  to the g l i d e  and  i s the  direction  c a l c u l a t e d . For  i s used.  A  set  the  of  system.  system  glide  perpendicular  i s to  direction  plane.  The  c a l c u l a t i o n s the T,  The  i n order  corresponding  quantities  project  with  the  second to  and shear  resolved  make  any  stress shear  definition respect  of  to  of a  354  coordinate the  system  q u a n t i t i e s T,  is a  tensor  transforms  i f under  M  defined  a)  The (P, over  6,  i s the operator positive  Stress  when  i s done  transformation  from  z) to a c a r t e s i a n  the z-axis  =  M.T.M"  performing  i n Cartesian  as shown  rotation  of the  1  the r o t a t i o n  i n the clockwise  of a x i s  Coordinates  a  cylindrical  ( x , y,  system  z) i s i d e n t i c a l  of  coordinates  to a  rotation-9  i n Figure.  e  matrix  for this  M  z  (-6)  transformation i s  =  cos6  -sine  0  sine  cos e  0  0  for  this  type  which i s  direction.  Z  The  system  as  T  where  a  of transformations  0  the inverse  1  operation i s  355  -0)  and  =  M  =  z<~ > e  M  z ( e )  therefore oxx  oxy  axz  oyx  oyy  ayz  axz  ayz  az z  =  X  which  cos6  -sine  0  sine  cose  0  0  0  1  pz  0  0  -  cos6  s i n9  0  -s i n 6  cos8  0  0  0  1  gives a  =  XX  °yy  ' =  °P °P  a °xy  +  sine  + a6  z sin  '  a  =  T  ! =  T  xz a  cose  yz  e cos e  Two  z-axis  z)  has  the  second  cos  2 e  p  ( O  " °e  P  s in6 Z  i n t h e ( 1 1 1 ) P l a n e and  Direction  rotations  the  2  cos6  Pz  b ) S a m p l e C a l c u l a t i o n o f t h e RSS [1101  sin 6  e  a  0  zz  cos 3  X  p  are performed  by an a n g l e  the x'-axis  a = 3 / 4 IT  coincident  rotation  = / 3 / 3 , t h e new  over  the f i r s t  the rotated with  the  system  : with  the  x'-axis  ( x " = x',  rotation  system  ( x ' , y ' , z' =  [110] d i r e c t i o n by  an  angle  y" , z " )  over  has  3  ; with  such  the  that  z"-axis  356 perpendicular stress  to  component  The rotation  the is  matrix  (111)  o  plane.  In  this  case  the  desired  .  for  the  i s performed,  combined  rotation  i s calculated  in  the  order  the  as  M , (3 ) M ( a )  M(o . 3 )  x  z  with 1  0  0  0  M ,(3) x  cos B  0  sin3  -sin(3  cos S  and  M (a) z  cos a  sin a  0  -s i n a  cos a  0  0  0  Multiplying rotation  both  matrices  of the tensor  to  obtain  the d e s i r e d  M(a,  a X  x"  (o  z  + a + a  Similar The for  calculations  performed each  case  - a  xx  xz  3 ) and  performing  „ component  the  is :  ) s i n a cos a s i n 3 + 2  xy  2  ( s i n a s i n 3 - c o s a sin3 ) + cos a  are  rotations are  yy  z  1  listed  cos 3  repeated and in  + o  yz  s i n a cos 3  f o r the  corresponding Table  VI.1 .  other  11  calculated  projections, component  357  Table  VI.1  Rotation  performed  f o r the c a l c u l a t i o n of the  RSS  ROTATIONS  |  Slip plane  |  Slip | direction |axis  111  Til  111  111  When replaced are  :  the in  1 |  2  angle  |axis  angle  110  z  3/4  Oil  x  TT/4  Z*  3  z"x"  101  y  TT/4  X '  3  x"y"  101  y  3/4  -3  x"y"  Oil  x  TT/4  Z'  110  z  TT/4  X'  Oil  x  -TT/4  101  y  110  TT  TT  x  |  (Calculated Component | |  1  x'  g  x"z"  3  y"z"  3  x"z"  z'  -(TT-3)  x"z"  -3/4 TT  Z*  -(TT-3)  X " Z "  z  -TT/4  x'  3  x"z"  Oil  x  -TT/4  z'  - 3  x"z"  101  y  X '  TT/2-3  x"z"  110  z  x '  - 3  x"z"  values  terms  of  of the  TT/4 -3/4  the  TT  cartesian  cylindrical  TT/2-  components components  of  stress  are  the  twelve  RSS  358  Table  VI.2  R e s o l v e d Shear slip  Stress  | P l a n e| D i r e c t i o n |  [110]  (111)  [Oil]  component  Resolved  i n the <110>  Shear  (111)  Stress  / 6 < - ( 0 - 0 ) ( c o s e - s i n 6 )+x (cos9-sin0)} —p 0 pz' b 2  2  O  . o 2 / 6 { - ( 0 s i n 0 + o „ c o s 6)+ 6 ° (0„-O )sin0cos9+T cos9 zz p 0 pz p  9  +  _  fl  [101]  [Til]  2  .—  2  / 6 {o c o 0 + a „ s i n 9-a +(o - a ) s i n 6 c o s 0 - a sin0) — p 0 zz ' p 0' pz b Q  2  / ? {o c o s 8 + o s i n q P 0  [101]  —  [Oil]  /6  2  n  6  0-a  2  2  {-(o s i n 0 + a c o s p 0 Q  +0 2  zz  zz  / 6 ~ { - (o - o ) c o s 0 - s i n g P o  [Oil]  /6~{-(a ~6  [101]  - ( a -a„)cos0sin0-T sin0} p 0 pz  0)+  [110]  2  n  + ( o -o„ ) c o s 0 s i n 0 - T cos0} p 0 pz 0 )+T  pz  (cos0-sin0)}  sin 0+a cos 0 ) + 2  2  Q  P  (111)  }  6  +o 2  / 6 ~ {o c o s 0 + a „ s i n ~6" P  2  zz  - ( 0 - O Q  P  o  )cos0sin0-T  Pz  cos0}  0-  6  -a  [110]  zz  2  /?T { - (Op -Og ) ( c o s 0 - s i n 6  +(o.-o ) c o s 6 s i n 0 + T . sin0 } P 0 Pz f l  2  0 )+x  p z  (cos0-sin0 ) }  Cont . /  359  Cont./ [Oil]  / ~ (-(o. s l n 6 + Q 2  Q  cos 0 ) + +0  (111)  [101]  /6~ (o cos 0 + a s i n ~6 Q  P  zz  /6" {-(o 6  p  -o„ ) cos0 sln0+T cos8 } 0 pz  0-  6  -a  [110]  +(a  z  - ( a - o ) cos0 sin©+T sin0 } p 0 pz n  - a ) ( c o s 0 - s l n 0 )-T (cos0+sin0)} P y P z 2  Q  2  360  Using elegant Table  form  VI.3  trigonometric as f o l l o w s Compact  |P1ane|Direction| [no]  (111)  [oTi]  relations  they  can  be  put  in a  :  form  Table VI.2  o f t h e RSS f r o m  Resolved /6~{-(0 ~6  Shear  -O P  /6~{-(a ~6~  Stress  | Mode |  ) c o s 2 6 +/2T c o s (8 r  a  6  P  -Or,  p  +TT/4  ) }  I a  C  ) / 2 s i n 6 s i n (8 + T T / 4 ) + ( a - 0 ) + r  6  ?  Q  6  +a _ c o s 8 }  PC  [ioT]  /6~ ( a -Or, ) / 2 c o s 8 s i n (8 + T T / 4 ) - ( a - o ) ~"g~ P D C o -T s i n 8 } r  r  [oTi]  a  111  a  IV  a  ~ { (o - o ) / 2 c o s 6 c'os (0+TT/4 ) - ( o - a ) p 0 C 8  —  Q  c  -x [ i n ]  11  Q  PC  [ToT]  more  /6"{(o 6  P  p C  sin0 }  - 0 ) / 2 s i n e c o s ( 0 + T r / 4 ) + (a Q  o  -T  Q  -a ) -  C o Q  V b  COS0 }  PC [no]  /~" { - ( O -Og ) c o s 2 0 + / 2 T ^ C O S (0 +TT/4 ) } 6  [Oil]  / 6 ~ {-(Op-Og ) / 2 s i n 0 s i n ( 0 + T r / 4 ) + (a^-Og ) 6 -T ^ COS0)  p  I a  p  p  (TTi)  [ToT]  / i ~ {(O 6  D  -OQ  b  111  a  )/2cos0sin(0+Tr/4)-(Or-Og ) + +T  [iTo]  11  / 6 ~ { - ( 0 -O ) COS20+/2T ~e p 6 PC  p C  sin0}  COS(0+TT/4)}  I a  Cont./  361  Cont . / [Oil]  /6~{(o 6  p  -0. ) / 2 s i n 6 c o s ( e + T r / 4 ) + (o -a.)  y  C o  +T  (111)  [ToT]  /6~{(0 6  P  [!!o]  _cos6}  V a  PC  -0 )/2cos6cos(e +TT/4)-(a -0 )  o Q  r  +T  Z i " { - ( 0 -0 ) c o s 2 9 - / 2 6 P 9  T  PC  +  r  Q  C o  sin8)  PC  COS(6  +  TT/4)}  + IV  b  I b  362  APPENDIX V I I  Temperature The  Field  problem  During to solve  3  1  33  I.C. s  3p  equations.  33  2  —  ) +  by t h e f o l l o w i n g  =  3^  (VII.l)  r  9t  at t * = 0  1  3  3p  i s given  33  ( p  p  Cooling  =  P  0  B.C. s  0  +  P  l^  +  P  2  ^  (VII.2)  at t * > 0  1  33 /  3p  =  -h'3  (VII.3)  =  -h'3  (VII.4)  =  h'3  (VII.5)  p= l  33 /  3?  C=Ct  33 /  A solution 3(p.  i s obtained £.  t*)  =  by s e p a r a t i o n o f v a r i a b l e s i . e .  R ( ) Z ( ) T*(t*) p  C  (VII.6)  -Tt* Morever, in  we  (VII.l)  assume T * ( t * ) a e and o p e r a t i n g  we g e t  , ¥ i s constant.  Substituting  3  363  1  1  d  R(P)  P  dp  (p  dR(p)  1  ) +  dp  d Z(C) 2  =  -f  (VII.7)  dC  Z(C)  2 which  is satisfied  respectively. equations  i f each  This  gives  term the  following  say  ordinary  -X  and  -y  differential  and c o n d i t i o n s  d R  1  2  dR  +  d(Xp) with  i s constant,  2  R such  + R  (Xp)  2  =  0  (VII.8)  d(Xp)  that  dR / dp  =  -h'R  Z  =  (VII.9)  p=l  and  d z 2  2  —  dC with  + y  0  (VI I.10)  -h'Z  ( V I I . 11 )  2  Z subject  to  dZ / d  £  = ?= ti  t  dZ d?  and  also  / = h ' Z ?=0  X, y  and T must  (VI I.12)  satisfy  364  ( V I I .13)  The  solutions  functions  of zero  satisfying  h'  J  to  i . e .R(p) equation  i n Appendix  solutions satisfy  and  (VII.9)  are  the  J ( X p ) f o r a l l the Q  (see Appendix  Bessel A values  VIII) ( V I I .14 )  (X)  Q  and o r t h o g o n a l  The order  order,  =  shown  complete  (VII.8)  the a l g e b r a i c  X J j (a)  As  of  VIII,  these  functions  constitute  a  set of v e c t o r s .  of  (VII.10)  t h e B.C.'s  are  (VII.11)  the  harmonic  and  (VII.12)  functions, we  in  choose  as  solutions  Z  (C)  =  substituting  A cosy  K + B siny  i n (VII.12)  A  Y  B  h  and  £  (VII.13)  (VII.15)  we  obtain respectively  that  (VII.16)  and 2Y h tan  Y C  T  (VII.17)  =  Y'  365 (VII.17) the  gives  the  functions  orthogonal The  3 ( p-  condition  (VII.15)  subject  (see Carslaw  general  1*)  =  and  solution  Z C*( a.y)  to  e  to  the summation extends  roots  of  To (VII.2)  obtain  the  (VII.12)  are  2  2  +  i s therefore  2 )t*  Y  J (Xp ) +  to a l l the A  (VII.17)  (VI I.18)  hr  sinyC]  and  y -values  which  are  respectively.  coefficients  i s a p p l i e d and t h e i n n e r  performed,  and  Moreover,  Q  where  and  y-values.  (VII.11)  [YcosyC  (VII.14)  the  Jaegger).  for 3  -(A  obtain  C*( \,y product  )  the  initial  condition  i n the space-function i s  that i s  fC t =  It  i s observed  C*(X,Y)  After  dp  C*(A,Y>  ( V I I . 19 )  J 0  that  =  performing  C (X) 1  C*(A,Y)  c a n be w r i t t e n a s (VII.20)  C (Y) 2  the i n t e g r a t i o n  the c o e f f i c i e n t s  are found  t o be  366  CAX)  =  (VII.21) (h  + A ) J (A)  , < J  Q  and  C ( ) 2  = 1 1 / 1  y  (VII.22)  with  't  (y  + h' )  2  + h'  2  (VII.23)  and  II  =  siny^  C  0  S  Y  +  4  t  {p  —  +  Q  (ti  P l  <-P h'  +  t  h ' / Y  2  + P (C  )  + P j ( l - h'C )  0  -  2  2  + P ( ?  2/Y  2  t  2  t  2  +  2  h'/j )}  +  2  t  2 2 - h'Cj + 2 h ' / Y ) } +  Y  P h'  p,  n  + Y  Also  2p h ^ —  1  the  Y  (VII.24)  Y  solution  independent  1  for 3  c  a  n  be  written  as  a  product  of  two  series  _A t* 2 h [ Z C ( A) e A 2  3( p. £ . t * )  =  1  1  J (Ap)] n  0  .Y t* 2  {Z Y  C 2  ( Y )  E  [ Y C ° S Y  £  +  h'siny?]}  (VII.25)  367 APPENDIX  Analytical  The  S o l u t i o n of the Temperature  partial  3 3  subject  differential  l  2  3p  VIII  p  2  96  3 3  3p  3£  Field  equation  33  2  2V  3C  2  to the f o l l o w i n g Boundary  t o be  =  solved  is :  0  (VII I .1)  Conditions  33 /  3p  Equation  p=  (VII.1)  3 ( p,C )  which  1  -hr 3  (a)  at  £  =  0  at  £  =  C  Q  c a n be  (b)  by  t o t h e f o l l o w i n g two  d R  1  2  2  separating  variables  + R Xp  ordinary  differential  equation  dR  + d(Ap)  (c)  t  solved  (VII I . 2 )  R(p ) Z(£ )  =  leads  =  d(Ap)  =  0  (VIII.3)  368  d R — d«T  dZ  2  - 2V  Solutions R(P)•J  - X Z  of  (VIII.3)  0  (VIII.4)  are the Bessel  functions  of  zero  order  (XP)  Q  Equation coefficients  (VIII.4) with  i s linear  the general  £ / /V V + 2  Z(C)  where  =  d£  =  A^  solution  (A  and  B^  + B  with  constant  solution  —  2  e>  x  and homogeneous  x  E VV V +A A 2  e ^  are constant  to  be  2  V£  ) e ^ V  determined.  The  general  i s therefore  6 (p,C)  =  e  V  ZJ (Xp)  C  Q  [A  e * k  A  C  + B  x  e  k  ^  (VIII.5)  5  X  2  2  where  = V  possible  values of  From Xj  from  J f  From  i s carried  out over  a l lthe  ( V I I I . 2 a ) we g e t t h a t  (X)  which  + A . and t h e summation  =  hr  Q  the values (VIII.2b)  J (X)  (VIII.6)  Q  of X are obtained.  and ( c ) i t i s o b t a i n e d  that  369  1  =  I A  [A  J (Ap) 0  x  B  +  ( V I 1 1 .7)  ]  A  and V  e  In is a  order  necessary  C  t  k  I Jg(Ap) A  complete  sub-set  To  this  expansion  1  0  p  d  0  where  that  C  +  e  l e t us  = z A  J 0  1 was s u b s t i t u t e d  t h e above  J(Ap ) f r o m  the  i n terms  <Ap) P  by  (VIII.8)  ]  equations i t (VIII.6)  form  functions.  calculate  function  - \ l t  from  the functions  of orthogonal  of the unit  J (k )  p  A t  t o c a l c u l a t e A-^ a n d t o show  do  [A^ e  J  (  k  0  (VIII.7)  P  )  coefficients  of J(kp),  d  P  (  A  that i s  V  + A  and p  f o r the  i s used  as a  weighing  the  integral  function. Using inside  t h e summation  for  A  J (Ap)J n  0  properties  0  *  the  Bessel  i n (VIII.9)  Functions  c a n be w r i t t e n a s  k  (kp) p d p  =  [kpJ (Ap ) J ( k p ) Q  A -k  0  2  A p j ( k p) J ( A p ) ] Q  of  Q  Q  2  —  Q  A"-k  [-kJ (A)J (k) Q  1  + AJ (k)J (A)] Q  1  370 1 [-J (X) h r J ( k ) Q  which  shows  were  obtained  J (Ap)  =  Q  the J (Ap)  + J (k) hr J (X)]  0  Q  using  the  following  h (Ap)pdp  2  \  =  2  (VIII.10)  the  from  J (Ap)'s  derivatives  gives  t h e norm  as  2 1—  are  n  (VIII.9),  we  j J  2 Q  +  B  A  (A  Another  2  + h r 2  equation  is valid  orthogonal  get a f t e r  2hr  This  between  equalities  (A)  (VIII.10)  X  Knowing  A  r  j l+  .0  A  two  t h e same p r o c e d u r e  u  Q  last  0  (VIII.6).  1  J  =  0  The  relation  1  A = k and  Q  are orthogonal.  Q  - J ( A p ) and E q u a t i o n  For  f  Q  norm  given  by  that  V I I . 11  )  J (A) Q  f o r A^ and  f o r any  integration  a  0 2  Q  with  p , which  is means  obtained each  term  from  (VIII.8)  should  cancel,  i.e.  A.  .  e  M  _ A t e k  t  +B,  ?  u  =  o  * Mathematical Methods i n P h y s i c s . W.A. B e n j a m i n E d s . N.Y.  (VIII.12)  J . Mathews  a n d R.L.  Walker.  371 From  (VII.11)  and ( V I I . 1 2 ) 2k  A  =  x  -B  x  e  we g e t  t%  X  (VIII.13)  t  and  0 2 2, . . + h r ) J ( A) 2  B  A  =  Finally  — 2 (A  the  temperature  h  r  "j 2k g , (1 - e A<=t )  h  Q  A  Q  complete  field  expression  c a n be w r i t t e n  vc  e(p,C)  1  solution  f o r the  V P> A  r  A A^  the  (VIII.14)  as  = 2hr e " Z  k  for  t  - e  (A 2 k  + I T r ^ ) J (A) Q  A^t  A? e k  (VIII.15) , 1 - e  with  k  2  A  = V  A Jj(A)  2  + A =  2  hr  2  k  A t C  a n d t h e A's  Q  J (A) Q  satisfy (VI11.16)  372  APPENDIX  Plane  Strain  The derived an  plane  strain  assuming  there  axial  axial  Analytical  displacement  t r a c t i o n at  In  the  IX  Stresses  analytical is in  radial  order  ends  non-dimensional  only  of  to  the  variables  3 Pd p  ,1  solutions  for  stresses  displacement  satisfy  the  and  are  allowing  condition  of  no  cylinder.  these  solutions  are  3pd p  (IX.1)  3pdp - 3  (IX.2)  rP  3p d p +  A  =  with  a  3 p d p - 3  2  conversion factor  a  d  A.  i  (VIII.15)  MP 1  m  Axisymmetric  For  E ( T  the the  Thermal  general stress  (IX.3)  given  " -  by  V  (IX.4)  V  Field  temperature  components,  after  field  given  integrating  by by  Equation  parts,  are  373  ve  op  =  o„  2hr  =  e ^ Z K ( £) X x  v e  K  ^ Z  °  U )  A  VC  ^  r  Jj(Ap  o  f-  2  X  h  0  +  [ J (X)  A  r  J (A)  Ap  X  {  )  (IX.5)  f  [  V  Q  2hr  9  h  n  r  V  Q  X  * Xp  J  n  (IX.6)  ( A p ) ] >  °  Ap)  (IX.7)  J (A) 0  where  K U>  (IX.8)  X  _ 2 k  (1  To  - e  o b t a i n these  > ^ i - ) ( A +2 A  fc  2 2  iTr*)  expressions the following  integral  was u s e d  ,1  J (Z)  ZdZ  Q  The  evaluation  accomplished  =  of  J (Z) Z  stresses  Q  at  the center  by u s i n g t h e f o l l o w i n g  J,  J (0)  (IX.9)  1  and  limiting  to  the cylinder  i s  values  (Ap)  /  1  Ap  1/2 p->-  0  (IX.10)  374  B.  Radial  Temperature  Field  a  temperature  field  are  immediately  o b t a i n e d as  For stress op  a  e  /  1  1 =  1  (  (3 P  2  1/2  -  by  3  p  the  values  of  : ( I X . 11)  1)  2  P  given  1)  (IX.12)  (IX.13)  375 APPENDIX  A N A L Y T I C A L AS I S Y M M E T R I C  X . 1  Axisymmetric  A)  Love's  The  2  with  V  V  L  , by  * L  symmetry,  potential equation  L  *  a  a o  cosine  A  n  dp  pdp  by  definition  =  finite  f c  {L  2 2  ,2 d  TT  , I  dp"  a  p  a p  p and  L  c  2  P  d  ,  n  F  fc  <•  a7  a  v  transform  on  2 2 TT  2~) £  L  (n,p)  c  t  relations  were  used  „ L  (n,p)  C  t  the  £ .  n (£,p)cos  2 * 9^L  a£'  n TT cos  TT£  d£  and a L*(£, )  satisfies  i  Fourier  pdp  the f o l l o w i n g  S  3  get  £~  ( £ , P ) >  d  of  £t F  L/EaS^r i o  :  only a function  e q u a t i o n we I d  where  STRESSES  Field  where  =0  Performing biharmonic  Love's  differential  2  S O L U T I O N S FOR  Potential  reduced  biharmonic  Temperature  X  £  d£  the  376  3L  3L  n  2 2 TT  (-1)' 3£  with  The  C  the assumption  t  c  L  ( n  -P  )  that  3L  3L  3C  3C  antitransform  T  3C  0  i s performed  by  summation  of  the  following  series  1 L  (Cp)  A fairly  general  modified  bi-Bessel  modified  Bessel  L  c  „ L  =  solution  =  =  nTT/^  °° Z  „  nrrC  L  (n.p)  of the biharmonic  equation,  functions  (n,p)  '  2 (0) +  i s obtained  of order  A ( l ) I (1 n o n  zero  cos  equation  which  as a c o m b i n a t i o n  i s the of the  and one as  ) + B ( l ) (1 n n  K  p)  1,(1 p ) 1 n K  with l  what  n  can  be  verified  using  the  following  among d e r i v a t i v e s o f t h e m o d i f i e d  Bessel  x  I' (x)  =  x I ^ U )  - x I (x) ;  x  I' (x)  =  x I  + x I ( x );  n  n  n  +  1  (x)  n  n  recurrent  functions  relations  377  The  functions The  are  general  not  form  included  of  the  because  they  d i v e r g e f o r p -»- 0.  a n t i t r a n s f o r m e d Love's  potential  is  therefore:  =  P)  L*U.  I  c £  [AUJ  .  n  n= 1  t  o  I  P)  (1  n  B ( l ) (1 n  +  p) n  1.(1  i  n  p)  cos  with  Aj  and n conditions. B)  c o n s t a n t s t o be  determined  from  the  boundary  Potential *  the  terms  of  a reduced  nondimensional  potential  can  equation 2 V  <b  *  =  be  p  accomplished  3 ( p , £ )  coordinates  distribution  i n the  function,  potential  quantities  cylindrical  this  body  a series  where and  of  a  2 3  g i v e n by  <b , £  by  V  2 = <J>( l - v ) / ( 1 +v) a S ^ r ,  and  3 ,  solving  is (p  the .  £  products  of  the  is  Goodier's  operator  the  one  variable,  as  a series  of  suggests  functions  t h a t we c a n * (b such that  propose  A V  Z  (£)  <b *  2  A  =  J ( A p ) Q  b e i n g the Z (£) x  =  Z (£) A  function e  [e  - e  e  ]  a  in  temperature  (VIII.15).  functions  and  differential  harmonic )  Equation  the  The (A,p)  J "  of  n  n  Goodier's  In  1 £  solution  form Z  of (£)  A  for  tb  *  378 Performing equation  a  sine  such  transform  2  *  V  3  Fourier  9 ?  the  differential  2  n  n-n  3£  0  * u) +  n+1  A  * r ^ — sin 1  1  A, n ,n  9<J)  i  on  that  2 fs  finite  £ d £ n  2  >  3?  t  where  <|> (p.C) s i n l  A.n  with  x  the a n t i t r a n s f o r m  4>  A  £ d £  by  (£.p)  £  We  given  n  n  get the transformed  1 3 < — • - — -  =  1  t  differential  equation  as  3  3p  Where  for  i t was  l / )  p 3p  assumed  that  3<1>  34>  3£  3£  a particular  solution.  (n.p)  =  J (Ap) 0  Z  (n)  379 Also Z,  A,  The  n  (n)  n  =  F  differential  o  {Z,  s  (£)}  A  equation  f o r (b.  n  A»  d  I  2  d  l —x) A  + d(Ap)  p d(Ap)  2  A particular  O'x.n  Substituting of  J^and  V  A  -  and  * (b, „ ( n . p ) ' X  ,  using  the following  J' (Ap)  =  -  Jj(Xp)  J" (Ap)  =  - J (Ap) +  and  0  JjUp)  n  Ap we  get  that  A,n< > i ,,2 +1 /A n  Z  \  A  2  n  2  and  z A  ^A.n  =  "  x  =  A  2  —  Z. A  ,  n n  (n)  as  A p )  :  0  0  n  c a n be o b t a i n e d  to get  J (Xp)  2  n  2  solution  c a n be m o d i f i e d  , ("> n  2 +  1  2  J  °  U  P  )  relations  among  derivatives  380  Applying  the  antitransform  ,  2  * <e.p>  =  A  Z  0  £  The  complete  solution  z  oo  - J Qp> —  2  n=l  t  f o r <J>  is  2  Z A  z  oo E  C  which  t  n=l  A (A 2  2  c a n be p u t i n a m o r e * 4) (p.£)  4hr  =  C  )  n  therefore  o  2  >  (  + h r ) 2  (1 -  2  e~  suitable  J (A) Q  2 k  way  A t) C  as  oo  Z n=0  €  t  sin 1  £*  J (Ap )  I  0  J (A)(A  h r  2  Q  2  +  C. C a l c u l a t i o n  2 Q  )(A  2  +  of Constants  l  ) ( l  2 n  Aj  The  non-dimensional  Goodier's  potentials  components  o (e.p) n  M  =  3£  "  2 k  A^t, )  and  Z , (n) A, n  Stresses  n of s t r e s s  derived  from  are 9  9  e  and n  and  + l  ni  /A )  2 n  2  (n) s i n 1 £  A .n  (1 + l  sin 1  n  (A  J  2hr.  4) (p.C)  (n)  '  [vv L  »  3 L dp  *  1 3  2 * 4)  —] + — ( — - — - B) l-v  3p  Love's  381  9 o (Cp)  1  t  =  e  9L  P  9p  [(2  - v) V L  3  After  P  +  [(1  2 * 9 L  - v) V L  to the following  x _ PC  =  derivating,  I  ( 1  i  2 * 3d>  1-v  3p3£  +  conditions  o -  p.= 1  X{< 0  ]  3£  boundary  15"  E  the stress  a  -  and  0  3 )  J  o *  3p  =  3 ) 3p  2  3£  o  3<b  2 * 2 * 1 3 <J) 3 L (• — 2 ] + 2 3£ 1-v 3£  3  Subject  1 ( 1-v p  ] +  3£  PC  1  tvV L  5,  £= o  components a r e  n > P  t  A  l  (2V  - 1)]+  n  n  1 t  +  B  l n > ( 1  - "  p  n  l p n  2hr 1  _  Z Q  A,n  ( n )  Q  ~ l^t  V  2k  (1  - e  tJ (Xp)  A  2 1„ A  2 2 *) ( A + h r ^  2  + -^(Apl/Apl  2 2 ) [1+(WA) ]J (X) Q  sin  1 £ n*  382  V£ n  2  C t  2hr  s  l  n  1  Kf v  C  Z  Q  ^  +  A  ' (A + h r  7 s i n 15 n  ( 1  n  (  Jn L _ 1 p n 1  P  PC  1  '  I n  «t  n  2 0  Z  \  V  [Aj  n  l > V n  2 (A%  A  V  c o s  2  h r  {< 0 v. I  ( 1  2 Q  n  n  nP  )  B  ^  n  l  p  +  n  2 n  /A )] 2  Ap  A(A^h r 2  2 0  *)  (1 +  n  1^/A ) 2  ]  Z  ~  l  A,n 2 R  {  A t )  2 ( 1 + 1„ A  C  A  l  ( n )  l  1  n  p  2 )  }  )  +  V >  I  A  A  - 2(2 - V , B  ) (1 - e  _  "  l  n  +  + 2 ( 1 - y ) Bj ]} l n  1  ] l 3 n  B j  (1 +  Q  Q  0  B  - J (Ap)  ) J ( A ) (1 - e  {(I (l P) v  p )  n  n  A  2hr +  o  V P>  n  [Aj  - i (l p)(2V-l)  A  2k  2  2hr +  )  (n) t J ^ A p ) /Ap 2  +  l  I  Z  ) J (A)d 0  A,n  ( n )  + l  2 n  /A )d 2  - e  }  383 These by  components  E a  |o| dim  3  of non-dimensional  to  obtain  = F S„| O I a f  J 4  the  stress  stress  in  limits  a r e used  be  absolute  . For the calculation  p = 0, t h e f o l l o w i n g  should  multiplied  units  of stresses  ;i.e.  at the axis,  :  J^O)  =  0  1/2 J (0) 0  Jj(x) 1/2  I (x) 0  x 0  x 0  The  evaluation  solving the  of the constants  the system  application  and is carried o u t by n n algebraic equations r e s u l t i n g from  of l i n e a r  A^  of the boundary  conditions  and  a  T  P =l 1 ,1 ( 1 n ) A  i  •]  B, [ I  +  l  (  l )  l  n  n  n  2  h  r  0  1-v  A  l W n  2 (n) ( l / + hr ) " °  1  Z  1n  3  +  1 -v  B  K.  A  l  [ I  1„  n  2  ,  0  n  ( 1  Z  „»  K  X  J  n  X.n  +  ( n )  2  (  1  "  v  )  !  l  (  1  n  )  ]  - ( 2 V - 1)  PC  =0 p =i  I (l )] 0  n  384 where  Z  >.n<"> -2k  A (A 2  Solving  f o r Aj  and  + h r  2  2  Bj  we  2 Q  )  (1 + l  2 n  /X )  [1 - e  2  £ A  t  ]  get  n 2hr = 2  ~  n  A(l-v)  — l  (Z K 2  (n)  ( I 0  X , n  n  n  ( l ) [1 n n  + 2hr  (l- )] v  0  1(1 +  1  ^  H  (hr  n  Q  + 2(l-v))  + 2hr  (1-v)]  }  and  1  B  2  h  2  r  - 7 <-"  l n  A  0  Z  n  K  2  X  "  (  n  )  J  "TT- V V [  1 - v  n  >  +  1  hr„ 0  n where  Z  i  2 'o'V  l  *  =  D.  Calculation  A.n<">  "  F  l  n  s  f  ~  of  <V  ^t  n>  t^ 2  2  Z  -v)  •  l  2 3  2 n  X, n  5,)  v£ e  ( 1  -k C  _  A  [e  - e  2  k  A t ?  A e  k  ?  n T T  ] sin  £ d  £  +  385  n  !  TT  A t  , ,,11 - (-1)  e  A t  t  -  , n (-1) e  2 2 (V-k )»  •  x  2.  A n a l y t i c a l Axisymmetric The  same  procedure  n  2 2  TT  n  (v+k )" +  TT  A  Solution  form  t>,  f o r (3 =  part  1  p  i s followed  applied  to a  2 temperature  A.  Love's  The same  field  differential  The  2 * 3d) + 2  2 * frn — + „ 2 3p  9  only,  i . e . 3 = -p .  f o r t h e Love's  I  (  1  P  )  +  B  (  P  )  equation f o r the Goodier's  * dd>  p  3p  finite  2 * 3<J) 3C  Fourier  0  0  8  1 n  n  . 3p  .2 - 1  Potential i s  2  2  transform i s performed,  * 8<{>  p  1  i s the  Potential  1  1  potential  A, a s w e l l a s t h e s o l u t i o n i . e . °° Z [A 0 n l n ^ ( ^ P * ! n=l n n  differential  a sine  on r a d i u s  equation  as shown i n P a r t 2 L*(£,p) = 4  3p  depends  Potential  B. G o o d i e r ' s  After  that  * 4> n *n  2 . -p f i H  we g e t  ^  386 where n  TT  sin  The  particular  gives,  after  £ d £  solution  solving  -  [1  proposed  is  <b n  (-D  ap  ]  + bp + c  ,  which  f o r a, b and c  4f 2  Finally,  0  '  performing the  antitransform  2 cb (p.£)  1 -  =  (1)" ~  I £^  C) C a l c u l a t i o n  ,  4 (p  1  1  of Stresses  and  5-)  +  sin 1 £  i  n  Constants Aj  and  Bj  n The  stress  potentials  components  derived  d  0  p )  n  [A  - B  n  A j  l  X  both  Love's  (  1  n  p  )  [  B  l  (  1  n  p  n  " 7^  )  n  (2  V  2  5*  Goodier's  V  D]  t l - ( - D  3  ] }  -  X  I  {  [  —l 2 n- _ (  X  1  „P  p  n  2  <—  +  n  —  and  are :  I { U n  +  from  n  +  > )> s i n 1 (  p  J  n  )  -  A n  !  ( 1  p )  ( 2 v - 1)] l  B n  3 3  n  +  387  [1  -<-l) ] ( l - v )  °e  ( { I I  Z  4  [1  1n  ( l  0  1n  p )  n  P  [Aj  )>  l nl  s  n  C  - 2(2 -v) B j ] n  n  I j d ^ ) Bj } l n  +  ; n  -(-1)"]  ' 1 J  ~„ 3  ( 1 - v )  S  l  n  1  n  i  l n  +  and 2 (  Z  {  0  I  (.  {  2  1  n P  [1  1 - v  These by  (  1  -  a  E  evaluation  )  B  I  l  (  1  n  p  p> c o s 1 '  2  n  non-dimensional  < >dim  n P  )  [  A  l  +  i)  A  l  o  components  the stress  [ I  n  0  ( 1  =  n>  0  "  of stresses  level  of the constants conditions  at  I l  p =  ! " 1 n 1  1  _  v  )  n n  l  B  l  ] }  +  should  i n absolute  be  units  multiplied ;  i.e.  * f <°>  a  the boundary  P  (  t%  Aj  and B j n  using  2  n  -(-I)"] 1  E a 3j. t o o b t a i n  The  )  >  ]  i s n  :  1  +  B  l  [  I  l n  (  1  n>  l  n  "  <  2v  "  l  )  done  as  before,  388  [1  - (-D  ] +  (1  1)  ) 1,  for  n = 2n  (2n + 1)TT  ii)  A  l  n  That  from  T  V C '  +  =  0  l  "  [I  n  at  *2  P  1  0 n*» C (1  non-zero  n  -  =  +  n  f o r A^  ^ 4 D(l-v) l  and B j  M  n  <W  "  2 ( 1  and  V )  come  n  Solving  l  - v ) 1,  i s the only  integer  A  B  (1  + 1) w i t h 1 = n  * i O  from  odd v a l u e s  n  and d r o p i n g  +  l  n  V  +2  the asterisk  W  and  n with  D ( 1 - v)  1 n  1 n  1 n  of the  389  1  (2n  =  + 1)TT  and  D  =  I  2 0  ( l ) n  l  n  -  [2(1 - v)  +  1„ ]  the  axis  2  n  The is  evaluation done  using  of  stress  the l i m i t  components values  given  at in  part  of  1 of  the I /P n  cylinder and  I /p  390  APPENDIX X I  COMPUTER PROGRAMS  XI.1  Mesh  Generator  c C  THIS  PROGRAM  GENERATES  MESHES  FOR  FLEMT  OF  BOULE  AND  PFLEMS  C C  FOR  CRYSTAL  , CONE  AND  SEED  C C DIMENSION  RC(4000),ZC(4000),NODE(10000,3)  DIMENSION  NV1(100),NV2(100),HV(100),TINFV(100)  DIMENSION  NT1(100),NT2(100).HT(1OO),TINFT(100)  DIMENSION  NH1(20),NH2(20),HH(20),TINFH(20)  DIMENSION  LASTN(100),M(100)  INTEGER  NN,NBC,NBCL,LAST,LASTEL,NEL0,NELP.NODE  INTEGER  CBN,LL,M,LASTN.BNS,BNL,NNHCV  INTEGER  NV1,NV2,NT  REAL REAL  1.NT2.NH1  ,NH2  XL,HBC.WBC,RC,ZC,HEIGHT,BH,RAD,BLE,BHS,BWL HV,HT,HH,TINFV,TINFH,TINFT  C C C  READ  DIMENSIONS  OF  CRYSTAL  AND  CONE  ANGLE  FOR  CYLINDER  THE  CRYSTAL  C RE A D ( 5 ,  1010)  READ(5,1015)  SIZE,XL SEEDW,TAN,SH  C C  CALCULATE  NUMBER  OF  BLOCKS  C NBC=XL/SIZE HBC=XL/NBC C C  CALCULATE  BLOCK  WIDTH  C NBCL=1  ./SIZE  WBC=1./NBCL C C  INITIALIZE  VARIABLES  C N N I N T =0 NEINT=0 ZINT=0.0 LASTEL=NEINT LAST=NNINT C C  GENERATE  FIRST  ROW  OF  NODES  C DO  50  1=1,NBCL  NOD=LAST  +  I  RC(NOD)=WBC*(1-1) ZC(N0D)=O.O 50  CONTINUE  C LAST=LAST+NBCL+1 RC(LAST)=1.0 ZC(LAST)=0.0 C C  GENERATE  REMAINING  C DO  200  1 = 1 ,NBC  DO  100  J=1,NBCL  C  ROWS  IN  391 NOD=LAST+J  100 C  RC(NOD)=WBC*(J-1) ZC(NOD)=HBC*I CONTINUE LAST=LAST+NBCL+1 RC(LAST)=1.0 Z C ( L A S T )=HBC*I  C C  GENERATE  CRYSTAL  TOPOLOGY  C DO 1 5 0 K= 1 , N B C L NEL0=2*K- 1 + LASTEL NODE(NELO,1)=LAST-2*NBCL+K-2 NODE(NEL0.2)=LAST-NBCL+K NODE(NELO,3)=NODE(NELO.2)-1 C NELP=2*K+LASTEL NODE(NELP,1)= NODE(NELO,1) NODE(NELP,2)=N0DE(NELP,1)+1 N O D E ( NE L P , 3 ) = N O D E ( N E L O , 2 ) C 150  CONTINUE LASTEL=LASTEL+2*NBCL  C C C 200 C C C C  CYCLE  FOR  NEXT  BLOCK  CONTINUE  GENERATE  NODES  AND  TOPOLOGY  FOR  CONE  M(1)=NBCL LASTN(1)=LAST C C C  CALCULATE  NUMBER  AND  DIMENSION  HE I G H T = ( 1 . - S E E D W ) * T A N CBN=HEIGHT/SIZE+1 BH=HEIGHT/CBN C DO 6 0 0 1 = 1 , C B N RAD = 1 . - B H * I / T A N L =I + 1 M(L)=RAD/SIZE+1 BLE=RAD/M(L) LASTN(L)=LASTN(I)+M(L)+1 C NLINE =M ( L ) DO 5 0 0 J=1,NLINE C  500 C  NOD=LASTN(I)+J RC(N0D)=BLE*(d-1 ZC(NOD)=BH*I+XL CONTINUE  )  RC(LASTN(L))=RAD ZC(LASTN(L))=BH*I+XL C DO  540  J=1,NLINE  C N E L 0 = 2*<J - 1 + L A S T E L NODE(NELO, 1) =L A S T N ( I ) - M ( I ) +J - 1 N O D E ( N E L O , 2 ) = L A S T N ( I ) + J+1 N0DE(NEL0,3)=N0DE(NEL0,2)-1 NELP=2*d+LASTEL NODE(NELP,1)=NODE(NELO, 1) NODE(NELP,2)=NODE(NELP, 1)+1 N0DE(NELP,3)=N0DE(NEL0,2) C 540  CONTINUE  OF  BLOCKS  392  IF(M(L).LT.M(I))GO  TO  549  C LASTEL=LASTEL+2*M(L) GO TO 600 C 549  C C C C 600 C C C  LASTEL=LASTEL+2*M(L)+1 NODE(LASTEL, 1 ) = LASTN(I )- 1 NODE(LASTEL,2)=LASTN(I) NODE(LASTEL,3)=LASTN(L)  CYCLE  FOR NEXT BLOCK  IN  THE CONE  CONTINUE GENERATE NODES  AND TOPOLOGY IN  THE SEED  BNS=SH/SIZE+1 BHS=SH/BNS BNL=SEEDW/SIZE+1 BWL=SEEDW/BNL LL=CBN+1 LAST=LASTN(LL) PHT=XL+HEIGHT C DO 700  I=1,BNS  DO 650  d=1,BNL  C C  650 C  NOD=LAST+d RC(N0D)=BWL*(J-1) ZC(NOD)=BHS*I+PHT CONTINUE LAST=LAST+BNL+1 RC(LAST)=SEEDW ZC(LAST)=BHS*I+PHT  C DO 680  J=1,BNL  C NEL0=2*J-1+LASTEL NODE(NE L O , 1 ) = L A S T - 2 * B N L + J - 2 NODE(NELO,2) = LAST-BNL+J N0DE(NEL0,3)=N0DE(NEL0,2)-1 C NELP=2*d+LASTEL N O D E ( N E L P , 1)=N0DE(NE L O , 1) N0DE(NELP,2)=NODE(NELP,1)+1 NODE(NELP,3)=N0DE(NEL0,2) C 680 C  CONTINUE LASTEL=LASTEL+2*BNL  C 70Q C "  CONTINUE NN^LAST NE=LASTEL  C C C  800  DETERMINE SEGMENTS WHERE CONVECTION OCCURS DO 800 1=1,NBC HV(I)=0.6 TINFV(I)=0.0 NV1(I)=(NBCL+1)*I+NNINT NV2(I)=(NBCL+1)*(1+1)+NNINT CONTINUE  393  810 C  820 C  830 C  DO 810 1=1,BNS I$=I+NBC HV(I$)=0.3 TINFV(I$)=0.0 NV1(I$)=LASTN(LL)+(BNL+1)*(I-1) NV2(I$)=NV1(I$)+BNL+1 CONTINUE DO 820 1=1,CBN HT(I)=0.S TINFT(I)=0.0 NT 1 ( I ) = L A S T N ( I ) d = I+1 NT2(I)=LASTN(d) CONTINUE DO 830 1=1,BNL HH(I)=0.0 TINFH(I)=0.0 NH1(I)=NN-I+1 NH2(I)=NN-I CONTINUE NNHCV=NBC+BNS NNHCT = CBN NNHCH=BNL  C C C  PRINT  THE DATA IN FORMAT TO BE READ BY FLEMT AND PFLEMS  W R I T E ( 6 , 1 0 0 5 ) NN,NE W R I T E ( G , 1 0 0 G ) NNHCV.NNHCH,NNHCT 1005 F0RMAT(5X.'NN= ' . I 4 , 5 X , ' N E = ',14,//) 1006 F O R M A T ( 5 X , ' N N H C V = ' , 1 3 , ' N N H C H = ' , 1 3 , ' N N H C T = ' , 1 3 , / ) DO 9 0 0 1=1,NN  900 C  910 C  920 C  930 C  940 C  950 C C C 1010 1015 1020 1025 1030 1040  Z C ( I ) = Z C ( I ) + ZINT W R I T E ( 6 , 1 0 2 0 ) 1 , R C ( I ) , ZC( I ) CONTINUE DO 9 1 0 K = 1 , N E WRITE(6, 1025)K,NODE(K, 1 ) , N 0 D E ( K , 2 ) , N 0 D E ( K , 3 ) CONTINUE NNCL=NBCL+1 T= 1 . 0 DO 920 1 = 1 ,NNCL WRITE(6, 1030)1,T CONTINUE DO 930 1=1,NNHCV W R I T E ( 6 , 1040)NV 1 ( 1 ) , N V 2 ( I ) , H V ( I ) , T I N F V ( I ) CONTINUE DO 940 1 = 1 .NNHCT WRITE(6,1040)NT1(1),NT2(I),HT(I),TINFT(I) CONTINUE DO 950 1=1,NNHCH WRITE(6,1040)NH1(I),NH2(I),HH(I),TINFH(I) CONTINUE FORMAT STATEMENTS FORMAT(2(2X,F10.5)) FORMAT(3(2X,F10.5)) FORMAT(2X, I 4 , 2 ( 2 X , F 1 0 . 5 ) ) F0RMAT(4(2X,14)) F0RMAT(2X,I3.2X.F10.5) F0RMAT(2(2X.I4).2(2X,F10.5)) STOP END  394  XI.2  Finite  Linear  Calculations  Element  Program  during  Growth  f o r the Temperature  C T H I S IS A PROGRAM TO C A L C U L A T E THE TEMPERATURE D I S T R I B U T I O N C IN L E C - G A A S DURING GROWTH USING A F I N I T E LINEAR ELEMENT C METHOD (SECOND V E R S I O N ) . C C DIMENSION I P ( 3 7 5 ) , R E S ( 3 7 5 ) I N T E G E R M LLB , LUB , I P , ITER , NHRS REAL*4 RES,EPS REAL *8 T B . D R A T I O INTEGERS LHB,NRHS.JEXP,NSCALE DIMENSION T B ( 2 ) INTEGER*4 NN,NDIMA,NSOL,NDIMBX,IPERM,NDIMT REAL*8 T M , R I N F , T , T T , D E T R E A L * 8 DTA , RBAR,A 1 , A 2 , A 3 , B 1 , B 2 , B 3 REA L *8 C1 , C 2 , C 3 , V K , T M $ , C O N S T , D E L 2 REAL*8 B L V , C O N S T V , C I N F V , B L T . C O N S T T REAL*8 CINVT,BLH.CONSTH,CINFH,RBARH DIMENSION T M ( 1 1 1 , 1 1 1 ) , T ( 1 1 1 ) , R I N F ( 1 1 1 ) , I P E R M ( 2 2 2 ) , T T ( 1 1 1 , 1 11) DIMENSION R C ( 5 0 0 ) , Z C ( 5 0 0 ) . N O D E ( 9 0 0 , 3 ) DIMENSION N T S ( 5 0 ) , H V ( 6 0 ) , T I N F V ( 6 0 ) , H H ( 6 0 ) , T I N F H ( 6 0 ) DIMENSION B L V ( 6 0 ) , N V 1 ( 6 0 ) , N V 2 ( 6 0 ) DIMENSION B L T ( 4 0 ) , N T 1 ( 4 0 ) , N T 2 ( 4 0 ) , H T ( 4 0 ) , T I N F T ( 4 0 ) DIMENSION B L H ( 4 0 ) , N H 1 ( 4 0 ) , N H 2 ( 4 0 ) , R B A R H ( 4 0 ) DIMENSION T M $ ( 3 , 3 ) , N ( 3 ) , R C $ ( 3 ) , Z C $ ( 3 ) DIMENSION T G ( 5 0 . 5 0 ) , T G R ( 1 1 1 ) , C V A L ( 2 0 ) , I 0 P ( 8 ) , V 0 P ( 8 ) REAL T G , C V A L , V O P , T M I N , D T E M , D X , D Y , C , X S I Z E , Y S I Z E INTEGER NX,NY,NRNG.IDIMX,NC,IOP REAL BORAXH,HBORAX,HARGON REAL T I N F A R , T I N F B O , Z B A R , T M B O R REAL TBOR,CA2.CA3,CA4,CB2,CBS,CB4 C C C C READ THE NUMBER OF NODES AND ELEMENTS OF THE SYSTEM C C READ(5,1005)NN,NE C C I N I T I A L I Z E PARAMETERS C C DO 10 1=1,NN RINF(I) = 0.0 RC(I) = 0.0 ZC(I) = 0.0 T(I) = O.O DO 10 J = 1,NN TM(I,J ) = 0.0 10 CONTINUE C C READ THE NUMBER OF NODES WITH S P E C I F I E D TEMPERATURES C AND THE NUMBER OF BOUNDARY S E G M E N T S . V E R T I C A L AND NON C V E R T I C A L , W H E R E HEAT CONVECTION OCCUR. C C RE AD ( 5 , 1010 )NNST , NNHCV , NNHCH , NNHCT C C READ GROWTH PARAMETERS:RAD I U S , C O N D U C T I V I T Y , V E L O C I T Y C AND L E N G T H . C READ(5,1015)R0,TK,V,XL,DTA C  READ(5,1018) CA,SW,SH,BORAXH READ(5.1019) CB2.CB3,CB4.CA2,CA3,CA4 C WRITE(6.1019) C C C  CB2,CB3.CB4.CA2,CA3,CA4  READ NUMBER OF NODE V S .  NODAL COORDINATES  DO 100 d = 1,NN READ(5 , 1 0 2 0 ) 1 . R C ( I ) , Z C ( I ) CONTINUE  100 C C READ ELEMENT NUMBER V S . NODE NUMBERS C DO 105 I = 1.NE READ(5,1O25)d.N0DE(d.1),N0DE(J,2),N0DE(d,3) 105 CONTINUE C C READ NODE NUMBER V S . S P E C I F I E D TEMPERATURE C DO 110 I = 1,NNST READ(5.1030) NT,TNT N T S ( I ) = NT T ( N T S ( I ) ) = TNT 110 CONTINUE C C READ PAIRS OF NODE NUMBERS D E F I N I N G BOUNDARY SEGMENT C WHERE CONVECTION O C C U R S , C A L C U L A T E THE CORRESPONDING C CHTC AND AMBIENT TEMPERATURE. C C TMBOR=1238.0 TMBOR=1.-(1238.-TMBOR)/DTA TBOR = T I N F B O ( D T A , T M B O R , C B 2 . C B 3 , C B 4 , 3 . 0 0 0 ) DO 140 I = 1 ,NNHCV REA0(5,1O4O) NV1(1),NV2(I) ZBAR=(ZC(NV1(I))+ZC(NV2(I)))*R0/2. IF(ZBAR.LT.3.OOO) TINFV(I)=TINFBO(DTA,TMBOR,CB2.CBS,CB4,ZBAR) IF(ZBAR.GT.3.000.OR.ZBAR.EO.3.000) TINFV(I)=TINFAR(DTA,TBOR,CA2. 1CA3.CA4.ZBAR) C I F ( Z B A R . LT . B O R A X H . O R . Z B A R . E O . B O R A X H ) HV(I)=HBORAX(DTA,TINFV(I)) IF(ZBAR.GT.BORAXH) HV(I)=HARG0N(DTA,TINFV(I)) C 140 CONTINUE C C DO 145 d=1,NNHCT READ(5,1040)NT1(d),NT2(d) ZBAR=(ZC(NT1(d))+ZC(NT2(d)))*R0/2. C IF(ZBAR.LT.3.000) TINFT(d)=TINFBO(DTA,TMBOR,CB2,CBS,CB4,ZBAR) IF(ZBAR.GT.3.000.0R.ZBAR.EO.S.OOO) TINFT(d)=TINFAR(DTA, 1 TBOR.CA2.CA3.CA4.ZBAR) C I F ( Z B A R . L T . B O R A X H ) HT(d)=HBORAX(DTA.TINFT ( d)) I F ( Z B A R G T . B O R A X H ) HT ( d ) = H A R G O N ( D T A , T I N F T ( d ) ) C 145  CONTINUE DO 150 d = 1,NNHCH REA0(5,1O4O) NH1(d),NH2(d) ZBAR=ZC(NH1(d))*R0 TINFH(d)=TINFAR(DTA,TBOR,CA2,CA3,CA4,ZBAR) HH(d ) =HARGON(DT A , T I N F H ( d ) )  C 150 CONTINUE C C WRITE HEADLINES C 155 WRITE(6,2010) WRITE(6,2020) WRITE(6,2030)  RO.V.XL NN,NE  396  WRITE(6,2035) WRITE(6,2038)  CA.SW.SH DTA,BORAXH  C C C C PRINT S P E C I F I E D TEMPERATURES C 175 WRITE(G,2080) DO 180 I = 1 ,NNST TSP= 1238. WRITE(6,2090) I , N T S ( I ) , T S P 180 CONTINUE C C PRINT BOUNDARY SEGMENT N O D E S , H T C H AND HTCV,AND C AMBIENT TEMPERATURE. C WRITE(6,2100) DO 190 I = 1 , NNHCVTINF=1238-(1. -TINFV(I))*DTA WRITE(6,2110) I . N V 1 ( I ) , N V 2 ( I ) , H V ( I ) ,TINF 190 CONTINUE C I F ( N N H C T . E O . 1 ) GO TO 196 WRITE(6,2105) DO 195 1=1,NNHCT TINF = 1 2 3 8 - ( 1 . - T I N F T ( I ) ) * D T A WRITE(6,2 1 1 0 ) 1 , N T 1 ( I ) , N T 2 ( I ) , H T ( I ) , T I N F 195 CONTINUE C C 196 I F ( N N H C V . E O . 1 ) GO TO 250 W R I T E ( 6 , 2 120) DO 200 1 = 1 , N N H C H TINF = 1 2 3 8 - ( 1 . - T I N F H ( I ) ) * D T A WRITE(6,2110) I.NH1(I),NH2(I),HH(I),TINF 200 CONTINUE C C CYCLE FOR EACH ELEMENT, K , AND FORM THE INFLUENCE C MATRIX OF THE SYSTEM C C 250 DO 400 K = 1,NE N1 = N O D E ( K , 1 ) N2 = N 0 D E ( K , 2 ) N3 = N O D E ( K , 3 ) RC$( 1 ) = R C ( N O D E ( K , 1 ) ) RC$(2) = RC(N0DE(K,2)) RC$(3) = RC(N0DE(K,3)) ZC$( 1 ) = Z C ( N O D E ( K , 1)) ZC$(2) = ZC(N0DE(K,2)) ZC$(3) = ZC(N0DE(K,3)) RBAR = ( R C $ ( 1 ) + R C $ ( 2 ) + R C $ ( 3 ) ) / 3 . A1 = R C $ ( 2 ) * Z C $ ( 3 ) - R C $ ( 3 ) * Z C $ ( 2 ) A2 = R C $ ( 3 ) * Z C $ ( 1 ) - R C $ ( 1 ) * Z C $ ( 3 ) A3 = R C $ ( 1 ) * Z C $ ( 2 ) - R C $ ( 2 ) * Z C $ ( 1 ) B1 = Z C $ ( 2 ) - Z C $ ( 3 ) B2 = Z C $ ( 3 ) - Z C $ ( 1 ) B3 = ZC$( 1 ) - Z C $ ( 2 ) C1 = R C $ ( 3 ) - R C $ ( 2 ) C2 = RC$( 1 ) - R C $ ( 3 ) C3 RC$(2) - RC$(1) =  DEL2= A B S ( R C $ ( 1 ) * ( Z C $ ( 2 ) - Z C $ ( 3 ) ) 1*(ZC$(1)-ZC$(2))) C C C  CALCULATE  THE ELEMENTS OF THE MATRIX  VK = ( 3 . 14 1 5 9 2 7 * V * R 0 ) / ( 1 2 . 0 * T K ) CONST = ( 3 . 1 4 1 5 9 2 7 * R B A R ) / D E L 2  + RC$(2)*(ZC$(3 )-ZC$(1 ) ) +  FOR EACH  ELEMENT  RC$(3)  397  c  C C C  T M $ ( 1 , , 1) 1 1 *VK T M $ ( 1 ,,2) 12*VK T M $ ( 2 , , 1) 1 1 *VK T M $ ( 2 ,, 2 ) 12*VK TM$( 1 3) , 13*VK T M $ ( 3 ,, 1 1 1*VK T M $ ( 2 , .3) 13*VK T M $ ( 3 , 2) 12*VK T M $ ( 3 , .3) 13*VK  == (B1 • • 2  + C1 • • 2 ) *CONST  == (B1 *B2 + C1 *C2) •CONST  R C $ ( 2 ) + R C $ ( 3 ) )*B  + (2.•RC$( 1 ) + + (RC$(1)  + 2. RC$(2) 4  == (B1 *B2  + C1 *C2) •CONST  == (B2 **2  C2 **2) •CONST  + (RC$(1)  + 2.*RC$(2)  == ( B l *B3 + CI • C 3 ) •CONST  + (RC$(1)  + RC$(2)  ) == (B1 *B3 + C1 *C3) •CONST  + (2 . *RC$( 1 ) +  == (B2 *B3 + C2 *C3 ) •CONST  + (RC$( 1 ) + R C $ ( 2 )  == (B2 *B3 + C2 *C3) •CONST  + (RC$(1)  +  + C3 • * 2 ) •CONST  + (RC$(1)  + RC$(2)  == (B3 •*2  + (2 . *RC%( 1 ) +  N ( 1 ) = N1 N ( 2 ) = N2 N ( 3 ) = N3 DO 300 1$ = 1,3 I = N(I$) DO 300 J $ = 1.3 J = N(d$) TM(I.d) = TM(I,«J) CONTINUE  RC$(2) + RC$(3))*B + R C $ ( 3 ) )^B  + 2 •RC$(3))^B  RC$(2) + RC$(3))*B + 2  2.*RC$(2)  ASSEMBLE SYSTEM MATRIX WITHOUT REGARDING BOUNDARY  + RC$(3))*B  •RC$(3))^B + R C $ ( 3 ) )*B  + 2 • R C $ ( 3 ) )*B  CONDITIONS  + TM$(I$,J$) 300 C CCYCLE FOR NEXT ELEMENT C 400 CONTINUE C C ACCOUNT FOR CONVECTION AT BOUNDARY,FORM RINF MATRIX C DO 420 I = 1,NNHCV BLV(I) = ABS(ZC(NV1(I)) - ZC(NV2(I))) CONSTV = 6. 1 8 5 3 8 3 2 * B L V ( I ) * H V ( I ) + R 0 R C ( N V 1 ( I ) ) TM(NV1(1),NV1(1)) = TM(NV1(1),NV1(1)) + CONSTV/3. TM(NV1(1),NV2(I)) = TM(NV1(1),NV2(I)) + CONSTV/6. T M ( N V 2 ( I ) , N V 1 ( I ) ) = T M ( N V 2 ( I ) , N V 1 ( I ) ) + CONSTV/6. TM(NV2(I),NV2(I ) ) = T M ( N V 2 ( I ) , N V 2 ( I ) ) + CONSTV/3. C CINFV = 3 . 1 4 1 6 H V ( I ) T I N F V ( I ) • R O ^ R C ( N V 1 ( I ) ) B L V ( I ) R I N F ( N V 1 ( I ) ) = R I N F ( N V 1 ( I ) ) + CINFV R I N F ( N V 2 ( I ) ) = R I N F ( N V 2 ( I ) ) + CINFV 420 CONTINUE C I F ( N N H C T . E O . 1 ) GO TO 435 DO 430 1=1,NNHCT B L T ( I ) = A B S ( S O R T ( ( R C ( N T 1 (I ) ) - R C ( N T 2 ( I ) ) ) * • 2 + ( Z C ( N T 1 ( I ) ) - Z C ( N T 2 ( I ) ) 1 )**2) ) RBAR = (RC(NT 1 ( I ) ) + R C ( N T 2 ( I ) ) ) / 2 . C0NSTT=3.1416 HT(I)•R0 BLT(I)/6. C TM(NT 1 ( I ) . N T 1 ( I ) ) = T M ( N T 1 ( I ) , N T 1 ( I ) ) + C O N S T T • ( 3 . * R C ( N T 1 ( I ) ) + 1RC(NT2(I))) C TM(NT 1 ( I ) , N T 2 ( I ) ) = T M ( N T 1 ( I ) , N T 2 ( I ) ) + C 0 N S T T * 2 . + R B A R C TM(NT2(I),NT 1(I))=TM(NT2(I),NT 1(I))+C0NSTT^2.•RBAR +  4  A  4  4  i  398  c  T M ( N T 2 ( I ) , N T 2 ( I ) ) = T M ( N T 2 ( I ) , N T 2 ( I ) ) + C O N S T T * ( R C ( N T 1(I ) ) + 1 3 . * RC(NT 2(1 ) ) )  C CINFT=3.1416*HT(I)*RO*BLT(I)*TINFT(I)/3. C RINF(NT1(I))=RINF(NT1(I))+ CINFT*(RC(NT 1 ( I ) ) + 2 . * R C ( N T 2 ( I ) ) ) C RINF(NT2(I ) ) = RINF(NT2(I ))+CINFT*(2.*RC(NT2(I))+RC(NT 1 ( I ) ) ) C 430 435  CONTINUE IF(NNHCH.EO.1) GO TO 445 DO 4 4 0 d = 1 ,NNHCH B L H ( d ) = S Q R T ( ( R C ( N H 1 ( d ) ) - R C ( N H 2 ( d ) ) ) * *2 + ( Z C ( N H 1 ( d ) ) 1d)))**2) RBARH(d) = ( R C ( N H 1 ( d ) ) + R C ( N H 2 ( d ) ) ) / 2 . CONSTH = 6 . 2 8 3 2 * R 0 * H H ( d ) * B L H ( d ) * R B A R H ( d ) TM(NH1(d),NH1(d)) = TM(NH1(d),NH1(d)) + CONSTH/3. TM(NH1(d),NH2(d)) = TM(NH1(d),NH2(d)) + C0N5TH/6. TM(NH2(d),NH1(d)) = TM(NH2(d).NH1(d)) + CONSTH/6. TM(NH2(d),NH2(d)) = TM(NH2(d),NH2(d)) + CONSTH/3. CINF = 3 . 1 4 1 6 * H H ( d ) * T I N F H ( d ) * R O * R B A R H ( d ) * B L H ( d ) R I N F ( N H 1 ( d ) ) = R I N F ( N H 1 ( d ) ) + CINF R I N F ( N H 2 ( d ) ) = R I N F ( N H 2 ( d ) ) + CINF CONTINUE  -  ZC(NH2(  440 C C MODIFY TM AND RINF TO ACCOUNT FOR NODES WITH S P E C I F I E D C TEMPERATURE C 445 DO 450 I = 1 ,NNST DO 448 d=1.NN RINF(d)=RINF(d)-TM(d,NTS(I)) TM(NTS(I) ,d)=0.0 TM(d,NTS(I))=0.0 448 CONTINUE RINF(NTS(I))=1. T M ( N T S ( I ) , N T S ( I ) ) = 1. 450 CONTINUE C C C C A L L SOUBROUTINE FSLE TO SOLVE THE LINEAR SYSTEM C OF EQUATIONS C 458 NDIMA = NN NSOL = 1 NDIMBX = NN NDIMT = NN CALL S L E ( N N , N D I M A , T M , N S O L , N D I M B X , R I N F , T , I P E R M , N D I M T , T T , D E T , d E X P ) GO TO 475 C C 4G0 LHB=21 IdMAX=(LHB-1)*NN+NN-(LHB-1) DO 465 1=1,IdMAX TB(I)=0.O 465 CONTINUE DRAT 10= 1 . E - 5 NRHS=1 NSCALE=0 DO 4 7 0 d=1,NN DO 4 7 0 I $ = 1 , L H B I=d+I$- 1 I F ( I . G T . N N ) GO TO 470 Id=(LHB-1)*(d-1)+I TB(Id)=TM(I,d) 470 CONTINUE C C CALL DFBAND C CALL D F B A N D ( T B , R I N F , N N , L H B , N R H S . D R A T 10 , 1 DET,dEXP,NSCALE)  u  475 WRITE(6,2500) DET,dEXP C C CPRINT OUTPUT C WRITE(G,2200) DO 500 I = 1,NN TGR(I)=1.238-(1.-T(I))*DTA/1000. WRITE(6,2300) I , T ( I ) , T G R ( I ) 500 CONTINUE C C C  CALCULATE  NEW GRID  NX = 50 NY = 50 IDIMX=NX C TMIN=TGR(1) C DO 550 I=1,NN IF(TMIN.GT.TGR(I))  550 C  WRITE(6,2400)  TMIN=TGR(I)  TMIN  C XMIN=0.0 YMIN=0.0 XMAX=10. YMAX=10. C DX= ( X M A X - X M I N ) / ( N X - 1 ) DY=(YMAX-YMIN)/(NY-1) C NRNG=2 C=10. 0PT=O.O C  600 C  DO 6 0 0 I=1,NN RC ( I ) = ( 1 . - R C ( I ) ) * 5 . 0 Z C ( I ) = Z C ( I )*5 . 0 CONTINUE CALL 1  C C C  PLOT  CGRID1(TG,IDI MX,NX,NY,XMIN,YMIN, DX,DY,RC,ZC.TGR.NN,C,NRNG,OPT)  GRID  XSIZE=10. YSIZE=10. NC=12 I0P(1)=1 I0P(2)=1 I0P(3)=0 I0P(4)=1 I0P(5)=1 V0P(5)=3. I0P(G)=1 I0P(7)=0 I0P(8)=1 V 0 P ( 8 ) = . 12 C DTEM=(1.238-TMIN)/(NC-1) C 700 C  DO 700 1=1,NC CVAL(I)=1.238-DTEM*(1-1) CALL 1  C  CONTUR(XSIZE,YSIZE,TG.IDIMX.NX.NY, CVAL.NC,IOP.VOP)  CALL PLOTND C C C FORMAT STATEMENTS C 1005 FORMAT(2(2X,13 ) ) 1010 F0RMAT(4(2X,I3)) 1015 F0RMAT(2X,F5.2,2X,F7.4,2X,F7.5,2(2X,F7.3)) 1018 F0RMAT(4F12.5) 1019 F0RMAT(6(2X,F10.5)) 1020 FORMAT(2X,I4,2(2X,F10.5)) 1025 F0RMAT(4(2X,14) ) 1030 F0RMAT(2X,13,2X,F10.5) 1040 FORMAT(2(2X,14)) 2010 FORMAT(1H1,2X,' A FLEM SOLUTION FOR THE TEMPERATURE F I E L D IN L E C - G 1AAS C R Y S T A L S ' , / / ) 2020 F0RMAT(5X,'RADIUS (CM) = ' , F7 . 5 . / / , 5 X , ' V E L O C I T Y (CM/SEC) = ' , F 7 . 5 , 1 / / 5 X , ' L E N G T H (CM) = ' , F 7 . 3 . / / ) 2030 FORMAT(5X. ' N O . OF NODES = ' , I 3 , / / , 5 X , ' N O . OF ELEMENTS = ' , 1 3 , / / ) 2035 F 0 R M A T ( 5 X , 'CONE ANGLE =' , F 1 2 . 5, ' D E G . ' . / , 1 5 X , ' S E E D WIDTH = ' , F 12 . 5, ' C M ' , / , 2 5 X . ' S E E D HEIGHT = ' . F 1 2 . 5, ' C M ' , / ) 2038 F 0 R M A T ( 5 X , ' T M P - T I N F =' , F 1 2 . 5 , ' D E G . C ' , / , 1 5 X , 'BORAX HEIGHT =' . F 1 2 . 5 . ' CM' , / ) 2040 FORMAT( 10X, 'NODES V S . NODAL C O O R D I N A T E S ' , / / , 6 X I ' , 13X . ' R ( I ) ' , 13X, l'Z(I)' .//) 2050 FORMAT(5X,I3,2(7X,F10.5)) 2060 FORMAT(' ' , / / / , 1 5 X , ' S Y S T E M T O P O L O G Y ' , / / , ' E L E M E N T NUMBER',11X,'NODE 1 NUMBERS' , / / ) 2070 F0RMAT(5X , 13, 1 0 X , 3 ( 5 X , 1 3 ) ) 2080 FORMAT( ' ' , / / / , 5 X , ' N O D E S WITH S P E C I F I E D TEMPERATURES',//.7X.'I',4X 1,'NODE',6X,'TEMPERATURE',//) 2090 FORMAT(2X,2(4X,I3),5X,F15.7) 2100 FORMAT( ' ' . / / , ' B O U N D A R Y SEGMENTS WHERE CONVECTIVE HEAT FLUX OCCUR' 1 , / / , 5 X , ' I ' , 6 X , 'NODE NUMBER PAIR ' , 4 X , ' H ' , 8 X , ' T I N F ' . / / , 2 0 X , ' V E R T I C A L 2 BOUNDARY' , / ) 2105 FORMAT(' ' . / / , 2 X , ' T I L T E D B O U N D A R Y ' , / / ) 2110 FORMAT* 2 X , 3 ( 3 X , 1 3 ) , 2 ( 8 X , F 10 . 5 ) ) 2120 FORMAT( ' ' , / , 2 0 X . 'NON-VERT ICAL B O U N D A R Y ' , / ) 2200 FORMAT( ' ' , / / , 5 X , 'NODE NUMBER', 1 0 X , ' T ( I ) ' , / / ) 2300 FORMAT( 10X , 1 3 . 6 X , 2 ( F 1 5 . 4 ) ) 2400 F O R M A T ( / / , 10X , 'TMIN= ' . F 1 5 . 6 ) 2500 FORMAT( 'DETERMINANT IS ' , G 1 6 . 7 , ' 1 0 * * ' , I 2 0 . / / ) STOP END C C C FUNCTIONS TO CALCULATE TINF C C REAL FUNCTION TINFBO(DTA,TMBOR,CB2,CB3,CB4,ZZ) REAL D T A , C B 2 . C B 3 , C B 4 . Z Z , T M B O R C TINFBO=TMBOR+(CB2*(ZZ-O.000)+CB3*(ZZ-O.000)**2+ 1CB4*(ZZ-0.000)**3)/DTA C RETURN END  401  c c  REAL REAL  FUNCTION T I N F A R ( D T A , T B O R , C A 2 , C A 3 , C A 4 , Z Z ) DTA,TBOR,CA2,CA3,CA4,ZZ  C TINFAR=TB0R+(CA2*(ZZ-3.0OO)+CA3*(ZZ-3.OOO)**2+ 1CA4*(ZZ-3.000)**3)/DTA C RETURN END C C C C C  FUNCTIONS TO CALCULATE HTC  REAL REAL  FUNCTION DTA,TE  HBORAX(DTA,TE)  C HBORAX=0.8*EXP(1 . 0 4 1 E - 3 * ( 1 1 . - D T A * ( 1 . - T E ) ) 13 . 267 1 E - 6 * ( 1 1 . - D T A * ( 1 . - T E ) )**2 ) C RETURN END C C REAL REAL  FUNCTION HARGON(DTA,TE ) DTA,TE  C HARGON=0.66*EXP(3.2319E-3*(11.-DTA C RETURN END  *(1.-TE)))  402  XI.3  Finite  Linear  Element  Program  f o r the Stress  Calculations  c C C C C C  T H I S PROGRAM CALCULATES THE DISPLACEMENTS AND S T R E S S E S IN L E C _ G A A S CRYSTALS DURING GROWTH USING A FLEM METHOD  REAL D V M S . X I M . X I , X C N REAL*4 STR,E$ DIMENSION S T R ( 6 ) , E $ ( 3 ) I NT EGER*4 I ERROR,NVEC REAL *8 E . A B DIMENSION A B ( 2 1 ) , E ( 6 ) INTEGER *4 N N . N E Q . L H . U INTEGER*4 JS . J S J S . U S , J S J I NT EGER* 4 N S O L , I P E R M . N S C A L E , L H B INTEGER*4 LLB,LUB,IP,JEXP,ITER INTEGER*4 NHRS RE AL *8 EPS.RES REAL*8 DAA,DRATIO DIMENSION D A A ( 4 0 0 0 0 ) REAL*8 DSS.DDET DIMENSION D S S ( 9 9 2 ) DIMENSION S I G R B $ ( 3 ) , S I G T B $ ( 3 ) , S I G Z B $ ( 3 ) , T A U B $ ( 3 ) , S S P $ ( G ) DIMENSION DSK$(6,6),A(3),B(3),C(3),DSS$(6) DIMENSION T$(3),N(3),RC$(3),ZC$(3) DIMENSION RC(600),ZC(600),NODE(1000,3),T(GOO),NSRD(100),NSAD(100) DIMENSION U ( 6 0 0 ) , V ( 6 0 0 ) , N S A S ( 1 0 0 ) DIMENSION U $ ( 3 ) , V $ ( 3 ) , S I G R $ ( 3 ) , S I G T H $ ( 3 ) , S I G Z $ ( 3 ) , T A U R Z $ ( 3 ) , X B A R ( 3 1 ) DIMENSION S I G R P ( 6 0 0 ) , S I G T H P ( 6 0 0 ) , S I G Z P ( 6 0 0 ) , T A U R Z P ( 6 0 0 ) , N E P N ( 6 0 0 ) DIMENSION D E L 2 P ( 1 0 0 0 ) DIMENSION N S S ( 3 0 0 ) DIMENSION W ( 7 ) , X L ( 3 , 7 ) R E A L * 8 DSK$1 , D S K $ , D S S $ REAL*8 F U N C , C 0 N S , A L P H A 1 , A L P H A 2 , B E T A 1,BETA2 REAL+8 E P R , E P Z , G R Z , T B A R , U , V REAL*8 XL,W,RBAR,DEL2,DEL2P,A,B,C C DIMENSION V M S ( 1 G G ) , V O N M S ( 5 0 , 5 0 ) , T E M P ( 5 0 , 5 0 ) DIMENSION C V A L ( 1 0 ) , I 0 P ( 8 ) , V 0 P ( 8 ) REAL V O N M S , T E M P , X M I N , X M A X , Y M I N , Y M A X REAL D X , D Y , V O P , C V A L , X S I Z E , Y S I Z E , S INTEGER IDIMX,IOP,NX,NY,NC,NRNG C C C  READ THE NUMBER  OF NODES AND NODES WITH S P E C I F I E D  READ(5,1005)NN,NE,NNSRD,NNSAD C C C  INITIALIZE NEO  VARIABLES  = 2*NN  C C LHB=26 LH=LHB-1 C UMAX = LH*(NEQ-1)+NEO C  8 C  DO 8 1 = 1 , U M A X DAA(I)=0.0 CONTINUE  DISPLACEMENT  DO 10 1 = 1 ,NEO DSS(I) = 0.0 C 10  20 C C C  CONTINUE DO 20 1 = 1 ,NN U(I) = 0.0 V(I) = 0.0 RC( I ) = 0..0 ZC(I) = 0.0 T(I) = 0.0 SIGRP(I) = 0.0 SIGTHP(I) = 0.0 SIGZP(I ) =0.0 TAURZP(I ) = 0 . 0 DEL2P(I)=0.0 CONTINUE READ CONSTANT  AND C O E F F I C I E N T S  R E A D ( 5 , 1 0 1 0 ) XNU READ(5,1015) RO.VEL.XLE.DTA C DO 100 J = 1,NN READ(5,1020) I , R C ( I ) , Z C ( I ) CONTINUE  100 C C READ SYSTEM TOPOLOGY C DO 105 I = 1,NE READ(5,1025)J,NODE(J, 1),NODE(J,2),NODE(J,3 ) 105 CONTINUE C C C READ NODE NUMBERS WITH S P E C I F I E D DISPLACEMENT C NNSRD=0 DO 110 I = 1,NN I F ( R C ( I ) . G T . 0 . 0 ) GO TO 110 NNSRD=NNSRD+1 NSRD(NNSRD ) = I W R I T E ! 6 , 1 0 3 0 ) NSRD(NNSRD) 110 CONTINUE C C C READ'TEMPERATURES  c  DO 120 <J = 1 , NN READ(5,1040)1,T(I) CONTINUE  120 C C PRINT DATA AND HEADLINES C WRITE(6,2010) WRITE(6.2020)R0,VEL.XLE WRITE(6,2030)XNU WRITE(6.2035)NN,NE.NNSRD C C C C  PRINT HEADLINES  FOR DISPLACEMENTS  WRITE(6,2100) C C C C C C C C C C  FORM MATRIX  GIVE  FOR I N I T I A L  STRAIN  AND S T I F F N E S S MATRIX  VALUES TO WEIGHT FUNCTIONS  COORDINATES  FOR NUMERICAL  OUINTIC ORDER  AND NATURAL  INTEGRATION OF  404  ALPHA 1 = . 0 5 9 7 1 5 8 7 1 7 8 9 7 7 0 BETA 1 = . 4 7 0 1 4 2 0 6 4 1051 15 ALPHA2= . 7 9 7 4 2 6 9 8 5 3 5 3 0 8 7 BE TA2 = . 101286507323456 W( 1)= . 2 2 5 W ( 2 ) = . 1 3 2 3 9 4 152788506 W(3)=W(2) W(4)=W(3) W ( 5 ) = . 125939180544827 W(6)=W(5) W(7)=W(6) XL( 1, 1 ) = 1 . / 3 . XL(1,2)=ALPHA1 XL(1,3)=BETA1 XL(1,4)=BETA1 XL(1,5)=ALPHA2 XL(1.6)=BETA2 XL(1.7)=BETA2 XL(2 , 1 ) = 1 . / 3 • X L ( 2 , 2 ) =BETA1 X L ( 2 , 3) =ALPHA1 XL( 2 , 4) =BETA 1 XL( 2 , 5) =BETA2 XL( 2 , 6 ) =ALPHA2 XL( 2 , 7) =BETA2  140  DO 140 1=1 7 XL(3,I)=1.• I)+XL(2,I)) (XL(1, CONTIMUE DO 400 K=1 NE N( 1 ) N(2) N(3)  NODE(K,1) NODE(K,2) NODE(K,3)  TBAR=(T(NODE(K,1))+T(NODE(K, 2 ) ) + T ( N O D E ( K , 3 ) ) ) / 3 . T $ ( 1 ) = TBAR T $ ( 2 ) = TBAR TBAR  T$(3)  RC$( 1 ) = RC(NODE(K, 1 ) ) RC$(2) = RC(N0DE(K,2)) RC$(3) = RC(N0DE(K,3)) ZC$( 1) ZC$(2) ZC$(3) A(1) A(2) A(3) B(1) B(2) B(3) C(1) C(2) C(3)  = ZC(N0DE(K, 1 ) ) = ZC(N0DE(K,2) ) = ZC(N0DE(K,3))  = RC$(2)*ZC$(3) = RC$(3)*ZC$(1) = RC$( 1 ) * Z C $ ( 2 ) ZC$(2) ZC$(3) ZC$( 1 )  RC$(3)*ZC$(2) RC$(1)*ZC$(3) RC$(2)*ZC$(1)  ZC$(3) ZC$(1) ZC$(2)  = R C $ ( 3 ) - RC$(2) = RC$( 1 ) - R C $ ( 3 ) = R C $ ( 2 ) - RC$( 1 )  RBAR = ( R C $ ( 1 ) + R C $ ( 2 ) + R C $ ( 3 ) ) / 3 . ZBAR = ( Z C $ ( 1 ) + Z C $ ( 2 ) + Z C $ ( 3 ) ) / 3 . DEL2 = A B S ( R C $ ( 1 ) * ( Z C $ ( 2 ) - Z C $ ( 3 ) ) + 1RC$(3)*(ZC$( 1 ) - ZC$(2) ) )  RC$(2)*(ZC$(3)  ZC$(1))+  405  c C C C  DO 200  IS= 1 .3  FORM MATRIX  FOR I N I T I A L  STRAIN  IN ELEMENT K  K$ = 2*1$ - 1 DSS$(K$)=T$(I$)*(B(I$)*RBAR+DEL2/3.)/2. C K$ = 2*1$ DSS$(K$) = T $ ( I $ ) * R B A R * C ( I $ ) / 2 . C C FORM S T I F F N E S S MATRIX FOR ELEMENT K C DO 200 J $ = 1,3 C C INTEGRATE NUMERICALLY TERM IN L ( I $ ) * L ( J $ ) / R 0 C FUNC=0.0 C DO 195 1=1,7 FUNC=FUNC+W(I)*XL(I$,I)*XL(J$,I)/ 1 (RC$( 1 ) * X L ( 1 , I ) + R C $ ( 2 ) * X L ( 2 , I ) + R C $ ( 3 ) * X L ( 3 , I ) ) 195 CONTINUE C I F ( I $ . E O . d $ ) GO TO 198 PIN=(1.-XNUj*DEL2/(24.*RBAR) GO TO 199 198 PIN=(1.-XNU)*DEL2/(12.*RBAR) C C 199 C  DSK$1=((1.-XNU)*B(I$)*B(J$)+(.5-XNU)*C(I$)*C(J$))*RBAR/(2.*DEL2) K$ = 2 * I $ - 1 L$=2*J$-1  C DSK$(K$,L$)=DSK$1+XNU*(B(I$)+B(d$))/6.+FUNC*DEL2*(1-XNU)/2. C K$=2*I$ C DSK$(K$,L$)=(XNU*B(J$)*C(I$)+(.5-XNU)*B(1$)*C(J$))* 1RBAR/(2.*DEL2)+XNU*C(I$)/6. C L$=2*J$ C D S K $ ( K $ , L $ ) = (( 1 . - X N U ) * C ( I $ ) * C ( d $ ) + ( 1RBAR/(2.*DEL2)  .5-XNU)*B(I$)*B(J$))*  C K$ = 2 * I $ - 1 C DSK$(K$,L$)=(XNU*B(I$)*C(U$)+(.5-XNU)*B(0$)*C(1$))* 1RBAR/(2.*DEL2)+XNU*C(J$)/6. C C C CYCLE FOR NEXT NODE C 200 CONTINUE C GO TO 262 C C C C A L C U L A T E EINGENVALUES OF THE C S T I F F N E S S ELEMENT MATRIX C C WRITE(6,2510)K C DO 250 I$= 1 , 6 DO 250 J $ = 1 , 1 $ I$J$=I$*(I$-1)/2+d$ AB(I$J$)=DSK$(I$,J$) 250 CONTINUE C  406 NVEC=0 NEQ$=6 CALL D 5 Y M A L ( A B , N E 0 $ , E , I E R R O R , N V E C ) C  260 C 261 C C C C C 262 C  DO 260 I$= 1 .6 WRITE(6,261) I $ . E ( I $ ) CONTINUE F0RMAT(I3,F12.8) W R I T E * 6 , 2 5 1 0 ) I ERROR  ASSEMBLE DO 300 I  MATRICES WITHOUT  REGARDING B . C .  I$ = 1 . 3  = N(1$)  C M$ = 2*1$ - 1 M = 2*1 - 1 DSS(M) = DSS(M)  + DSS$(M$)  C M$ = 2*1$ M = 2*1 DSS(M) = DSS(M) + D S S $ ( M $ ) C C DO 300  d$ =  1,3  C J  = N(J$)  C M$ = 2*1$ - 1 M = 2*1 - 1 N$ = 2*d$ - 1 N1 = 2 * J - 1 I F ( N 1 . G T . M ) GO TO 280 C Id=LH*(N1-1)+M C D A A ( I d ) = DA A ( 1 d ) + D S K $ ( M $ , N $ ) C C 280  N$ = 2*d$ N1 = 2 * d IF(N1.GT.M)  GO TO 285  C I J = L H * ( N 1 - 1 )+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C C 285  M$ = 2*1$ M = 2*1 IF(N1.GT.M)  GO TO 290  C IJ=LH*(N1-1)+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C C 290  N$ = 2*d$ - 1 N1 = 2*d - 1 I F ( N 1 . G T . M ) GO TO 300  C Id = L H * ( N 1 - 1 )+M C DAA(Id)=DAA(Id)+DSK$(M$,N$) C • C C ASSEMBLE ELEMENTS OF THE NEXT NODE C 300 CONTINUE  IN ELEMENT K  c  C CYCLE FOR NEXT ELEMENT C 400 CONTINUE C C MODIFY SK AND SS TO ACCOUNT FOR C C NODES WITH S P E C I F I E D DISPLACEMENT C C C FOR A X I A L NODES NO RADIAL C , DISPLACEMENT C DO 500 d$ =1,NNSRD dS=2*NSRD(d$)-1 d S d S = L H * ( J S - 1 ) + JS DAA(JSJS)=1 .0 DSS(JS)=0.0 DO 490 I $ = 1 , L H I=JS+I$ IJS=LH*(JS-1)+I DAA(I J S ) = 0 . O C 485 I F ( I $ . G T . J S . O R . I $ . E Q . J S ) GO TO 490 J=dS-I$ dSd=LH*(d-1)+JS DAA(dSd)=0.0 490 CONTINUE 500 CONTINUE C C C FOR NODE 1,N0 A X I A L C DISPLACEMENT C JS = 2 d S J S = L H * ( J S - 1 ) + JS D A A ( J S JS ) = 1 . O DSS(JS)=0.0 C DO 505 I $ = 1 , L H I=JS+I$ IdS=LH*(dS-1)+I DAA(IdS)=0.0 I F ( 1 $ . G T . d S . O R . 1 $ . E Q . J S ) GO TO 505 J=JS-I$ JS«J = L H * ( d - 1 ) + JS DAA(dSd)=0.0 505 CONTINUE C C DRATIO=1.E-5 NRHS=1 NSCALE=0 C C SOLVE THE LINEAR SYSTEM C C C A L L I N G DFBAND C C C CALL D F B A N D ( D A A . D S S . N E O , L H B , N R H S , D R A T I O , D D E T , d E X P , N S C A L E ) C WRITE(6,2005) DDET.dEXP C C C PRINT DISPLACEMENTS C C WRITE(6,2115) DO SOO I = 1,NN d = 2*1 - 1 U(I) = DSS(d) K = 2*1 V( I )=DSS(K)  408  c  WRITE(6.2110)1,U(I),V(I) 600 CONTINUE C C C C A L C U L A T E STRESSES AT A L L NODES IN EACH ELEMENT C C DO 700 I = 1,NE C N( 1)=NODE( 1 , 1 ) ' N(2)=N0DE(I,2) N(3)=N0DE(I.3) C RC$( 1 ) = RC(NODE(I . 1 ) ) RC$(2) = RC(NODE(1,2)) RC$(3) = RC(NODE(1,3)) C ZC$( 1 ) = Z C ( N O D E ( I , 1 ) ) ZC$(2) = ZC(NODE(1,2)) ZC$(3) = ZC(N0DE(I,3)) C C T B A R = ( T ( NODE ( I , 1 ) ) + T (NODE ( I , 2 ) ) + T (NODE ( I . 3 ) ) ) / 3 . C C T$( 1 ) == TBAR T $ ( 2 ) == TBAR T $ ( 3 ) == TBAR C U $ ( 1 ) == U ( N O D E ( I , 1 ) ) U $ ( 2 ) == U ( N O D E ( I , 2 ) ) U $ ( 3 ) == U ( N O D E ( I , 3 ) ) C V$( 1 ) == V ( N O D E ( I , 1 ) ) V $ ( 2 ) == V(NODE( I , 2) ) V $ ( 3 ) == V ( N O D E ( I , 3 ) )  c  A( 1 ) = R C $ ( 2 ) * Z C $ ( 3 ) - R C $ ( 3 ) * Z C $ ( 2 ) A(2)=RC$(3)*ZC$(1)-RC$(1)*ZC$(3) A( 3 ) = R C $ ( 1 ) * Z C $ ( 2 ) - R C $ ( 2 ) * Z C $ ( 1 )  C B(1) B(2) B(3)  == Z C $ ( 2 ) •== Z C $ ( 3 ) •== ZC$( 1 ) •-  ZC$(3) ZC$( 1 ) ZC$(2)  C C( 1 ) == R C $ ( 3 ) •- R C $ ( 2 ) C ( 2 ) == RC$( 1 ) -- R C $ ( 3 ) C ( 3 ) == R C $ ( 2 ) -- RC$( 1 ) C DEL2 = A B S ( R C $ ( 1 ) * ( Z C $ ( 2 ) - Z C $ ( 3 ) ) + R C $ ( 2 ) * ( Z C $ ( 3 ) - Z C $ ( 1 ) ) + R C $ ( 3 ) * ( Z C $ 1(1)-ZC$(2))) C RBAR = (RC$( 1 ) + R C $ ( 2 ) + R C $ ( 3 ) ) / 3 . ZBAR=(ZC$(1 ) + ZC$(2) + Z C $ ( 3 ) ) / 3 . C C EPR=(B( 1 )*U$( 1 ) + B ( 2 ) * U $ ( 2 ) + B ( 3 ) * U $ ( 3 ) ) / D E L 2 C E P 2 = ( C ( 1 )*V$( 1 ) + C ( 2 ) * V $ ( 2 ) + C ( 3 ) * V $ ( 3 ) ) / D E L 2 C GRZ = ( C ( 1 )*U$( 1 ) + C ( 2 ) * U $ ( 2 ) + C ( 3 ) * U $ ( 3 ) + 1B( 1 )*V$( 1 ) + B ( 2 ) * V $ ( 2 ) + B ( 3 ) * V $ ( 3 ) ) / D E L 2 C C C  CALCULATE DO 9 0 0  S T R E S S E S AT EACH NODE IN ELEMENT I$=1  I  ,3  C IF(RC$(I$).E0.O.O)G0  TO 895  C S I G R $ ( ! $ ) = ( 1 .-XNU)*EPR+XNU*EPZ+XNU*U$(I$)/RC$( I $ ) - T $ ( 1 $ )  c  SIGTH$(1$)=XNU*(EPR + EPZ) + ( 1 .-XNU)*U$(I  $)/RC$(1$)-T$(1$)  C SIGZ$(I$)=XNU*EPR+(1.-XNU)*EPZ+XNU*U$(I$)/RC$(I$)-T$(I$) C TAURZ$( 1$ ) = ( 1 . - 2 . * X N U ) * G R Z / 2 . C GO TO 899 C 895 C  SIGR$(I$)=EPR+XNU*EPZ-T$(1$) SIGTH$(I$)=SIGR$(1$)  C S I G Z $ ( I $ ) = 2.*XNU*EPR+(1 . - X N U ) * E P Z - T $ ( I $ ) C TAURZ$(!$)=(1.-2.*XNU)*GRZ/2. C 899  S I G R P ( N ( 1 $ ) ) = SIGRP(N(1$)) + SIGR$(I $)*DEL2 S I G T H P ( N ( 1 $ ) ) = S I G T H P ( N ( 1 $ ) ) + S I G T H $ ( I $ ) *DEL2 SIGZP(N(1$))=SIGZP(N(1$))+SIGZ$(I$)*DEL2 TAURZP(N(1$))=TAURZP(N(I$) )+TAURZ$(I$)*DEL2 D E L 2 P ( N ( 1 $ ) ) = D E L 2 P ( N ( 1 $ ) )+DEL2  C C CYCLE FOR NEXT NODE IN ELEMENT I C 900 CONTINUE C C C C C Y C L E FOR NEXT ELEMENT C 700 CONTINUE C C C A L C U L A T I O N OF AVERAGE S T R E S S E S AT A L L NODES C WRITE(6,2550) WRITE(6,2555) C C DO 960 I = 1 , N N C SIGRAV=SIGRP(I)/DEL2P( I ) SIGTHA = S I G T H P ( I ) / D E L 2 P ( I ) SIGZAV = S I G Z P ( I ) / D E L 2 P ( I ) TAURZA=TAURZP(I)/DEL2P(I) C C C A L C U L A T E P R I N C I P A L S T R E S S E S AND VON MISES S T R E S S E S C DO 950 I$=1 .6 STR(I$)=0.0 950 CONTINUE C STR(1)=SIGRAV STR(3)=SIGTHA S T R ( 4 )=TAURZA STR(6)=SIGZAV C N$ = 3 NVEC=0 C CALL S Y M A L ( S T R , N $ , E $ , I E R R O R , N V E C ) C VMS(I)=SQRT(((E$(3)-E$(2))**2+(E$(3)-E$(1))**2+ 1(E$(2)-E$( 1))**2)/2. ) C VMS(I)=VMS(I)*2.91*DTA WRITE(6,2560)1,SIGRAV.SIGTHA,SIGZAV,TAURZA,VMS(I ) C 960 CONTINUE C C PLOT CONTOURS OF VONMS AND TEMP  NC=10 NX = 50 NY = 50 IDIMX=NX XMIN=0.0 YMIN=0.0 XMAX=10. YMAX =10. DX=(XMAX-XMIN)/(NX-1) DY=(YMAX-YMIN)/(NY-1)  v  C DO 9 7 0 1=1,NN R C ( I ) = R C ( I )*5 . 0 ZC(I)=ZC(I)*5.0 970 C C  CONTINUE  VMSMIN=VMS(1) C 975 C  DO 975 1=1,NN IF(VMSMIN.GT.VMS(I))  VMSMIN=VMS(I)  VMSMAX=VMS(1) C 978 C  DO 978 1=1,NN IF ( V M S M A X . L T . V M S ( I ) ) VMSMAX = VMS(I ) DVMS=( VMSMAX-VMSMIN)  C NRNG = 2 S=10. OPT=0.0 C CALL CGRID1(VONMS,IDIMX,NX,NY,XMIN,YMIN.DX.DY, 1 RC . ZC , VMS . NN, S , NRNG , OPT ) C C XSIZE=10. YSIZE=10. IOP(1)=1 I0P(2)=1 I0P(3)=0 I0P(4)=1 I0P(5)=0 I0P(6) = 1 I0P(7)=0 I0P(8) = 1 V0P(8)=.08 IDIMX=NX C M= 1 XNC=NC C  985  DO 985 I = 1 , N C XI = I X1M=(XI-1 . ) / ( X N C - 1 . ) CVAL(I)=VMSMIN+DVMS*XIM**M CONTINUE CALL C0NTUR(XSIZE,YSIZE,VONMS,IDIMX,NX,NY, 1CVAL,NC,IOP,VOP)  C C CALL PLOTND 990 WRITE(6,2001) C C C READEING FORMATS C 1005 F0RMAT(4(2X,13)) 1015 F0RMAT(2X,F5.2,2X,F7.5,2(2X,F7.3)) 1010 FORMAT(2X,F5.3,2(2X,E12.5),2X,F10.2) 1020 F0RMAT(2X.I4,2(2X,F1O,5))  1025 F0RMAT(4(2X,14)) 1030 FORMAT(2X,I 3 ) 1040 FORMAT*10X,I3.6X.F15.4) C C WRITING FORMATS C 2001 FORMAT('SOLUTION F A I L E D ' ) 2005 FORMAT( 'DETERMINANT I S ' , G 1 6 . 7 , ' * 1 0 * * ' , I 20) 2010 F 0 R M A T ( 2 X . ' A FLEM SOLUTION FOR THERMAL D I S P L A C E M E N T S ' , / / , 5 X . ' A N D S 1TRESSES IN L E C - G A A S C R Y S T A L S ' , / / ) 2020 F 0 R M A T ( 5 X . 'RADIUS (CM) = ' , F 7 . 5 , / , 5X, ' V E L O C I T Y ( C M / S E C ) ='.F7.5,/, 1 5 X , ' L E N G T H (CM) = ' , F 7 . 3 , / / ) 2030 FORMAT(5X,'NU = ' , F 5 . 3 , / ) 2035 F 0 R M A T ( 5 X , ' N U M B E R OF NODES = ' , I 3 , / , 5X , 'NUMBER OF ELEMENTS = '.13 1 , / , 5 X , ' N O . O F NODES WITH NO RADIAL DISPLACEMENT = ' , 1 3 , / , 5 X , ' N O . O F 2N0DES WITH NO AXIAL DISPLACEMNT = ',13,/) 2040 F O R M A T ( 1 0 X , ' N O D E S VS NODAL COORDINATES AND VS T E M P E R A T U R E ' , / / , 6 X , ' 11 ' . 13X, ' R ( I ) ' , 9X , ' Z ( I ) ' , 4X , 'TEMEPERATURE ' , / / ) 2050 FORMAT(5X,I3,4X,2(3X,F10.5),2X,F15.4) 2060 FORMAT(' ' , / / , 15X,'SYSTEM T O P O L O G Y ' , / / , ' E L E M E N T NUMBER',11X,'NODE 1NUMBERS'.//) 2070 F0RMAT(5X.13,10X,3(5X,13)) 2080 FORMAT(' ' . / / , 5 X , ' N O D E S ON THE AXIS WITH NO RADIAL D I S P L A C E M E N T ' , / 1) 2085 FORMAT( ' ' , / , 5 X , ' N O D E S WITH NO A X I A L D I S P L A C E M E N T ' , / ) 2090 FORMAT(20X,13) 2099 F O R M A T ( / , 5 X , ' I T E R A T I O N NUMBER = ',13,//) 2100 FORMAT( ' ' , / / , 10X, 'DISPLACEMENT F I E L D : RADIAL AND AX I A L ' , / , 5X , ' NOD 1E NUMBER' , 4 X , ' R A D I A L ' . 11X, 'AX I A L ' , / ) 2110 F0RMAT(9X,I 3 , 2 ( 5 X , E 1 5 . 6 ) ) 2115 F 0 R M A T ( 5 X , ' C A L C U L A T E D DISPLACEMENT F I E L D ' , / ) 2120 F 0 R M A T ( / , 5 X . 'NODES AND DISPLACEMENTS AT THE FREE B O U N D A R Y ' , / ) 2500 FORMAT(' ' , / / , 2 X , ' S T R E S S E S CALCULATED AT ALL NODES IN EACH ELEMENT 1',//,2X,'ELEMENT',1X,'NODE',4X,'SIGMAR',4X,'SIGMATH',4X,'SIGMAZZ', 25X,'TAURZ',//) 2510 FORMAT(5X,13) 2520 FORMAT(1OX,I3,5X,4(E14.5)) 2550 FORMAT( ' ' , / , 5 X , ' A V E R A G E STRESSES AT EACH N O D E ' , / ) 2555 FORMAT(2X, ' N O D E ' , 2 X , ' N E P N ' , 7 X . 'SIGRAV ' , 7 X , ' S I G T H A V ' , 6 X , ' S I G Z A V ' , 8 X 1.'TAURZAV',/) 2560 FORMAT(1X,I 3 , 3 X , 5 ( E 1 4 . 5 ) ) C C STOP END  412  XI.4  Finite  Quadratic  Element  Program  f o r the Stress  Calculations  c c C C C C C C C C C  THIS PROGRAM CALCULATES THE DISPLACEMENTS AND S T R E S S E S IN L E C _ G A A S CRYSTALS DURING GROWTH USING A FOEM METHOD  ELEMENTS ARE CALCULATED  EXACTLY  AND USING  A OUINTIC NUMERICAL INTEGRATION FORMULA  REAL*4 STR,E$,VMS DIMENSION S T R ( 6 ) , E $ ( 3 ) I N T E G E R M I ERROR , NVEC . NEO$ REAL*4 E DIMENSION A B ( 7 8 ) , E ( 1 2 ) INTEGER*4 NN,NEO INTEGER*4 N S O L . I P E R M I N T E G E R M LLB , LUB , NHRS , I P , dEXP , ITER REAL*4 A , B , E P S , R E S DIMENSION AA( 1 1 1 6 0 ) , B B ( 9 0 ) , I P ( 9 0 ) , R E S ( 9 0 ) DIMENSION I d ( 9 0 , 9 0 ) REAL*4 S K , S S , D E S , D E S D D , D E T , S S N , D D E S , D E L D E S DIMENSION S K ( 9 0 , 9 0 ) , D E S ( 9 0 ) , S S ( 9 0 ) , I P E R M ( 1 8 0 ) „ D E S D D ( 9 0 , 9 0 ) DIMENSION SK$(12,12),SS$(12),T$(6),N(6),RC$(6),ZC$(6) DIMENSION A ( 3 ) , B ( 3 ) , C ( 3 ) DIMENSION R C ( 3 0 0 ) , Z C ( 3 0 0 ) . N O D E ( 6 0 0 , 6 ) , T ( 3 0 0 ) , N S R D ( 1 0 0 ) , N S A D ( 1 0 0 ) DIMENSION U ( 3 0 0 ) , V ( 3 0 0 ) . N S A S ( 3 0 0 ) DIMENSION U $ ( 6 ) , V $ ( 6 ) DIMENSION S I G R P ( 3 0 0 ) , S I G T H P ( 3 0 0 ) , S I G Z P ( 3 0 0 ) , T A U R Z P ( 3 0 0 ) , N E P N ( 3 0 0 ) DIMENSION D E L 2 P ( 3 0 0 ) DIMENSION N S S O O O ) C DIMENSION W ( 7 ) , X L ( 3 , 7 ) R E A L M ALPHA 1 , A L P H A 2 , B E T A 1 , B E T A 2 , W , X L C C C C C C C C  READ NUMBER OF NODES AND NODES WITH S P E C I F I E D DISPLACEMENTS  READ(5,1005)NN,NE.NNSRD,NNSAD C C C  I N I T I A L I Z E VARIABLES NEO  = 2*NN  C DO 10 1 = 1, NEO S S ( I ) = o.o DES( I ) = 0 . 0 C  10  DO 10 J = 1 , N E Q SK(I , J ) = 0 . 0 CONTINUE DO 20 1=1,NN U( I ) = 0 . 0 V(I) = 0.0 RC(I) = 0.0 ZC(I) = 0.0 T(I) = 0.0 SIGRP(I) = 0.0 SIGTHP(I ) = 0 . 0  20 C C C  SIGZP(I) = 0.0 TAURZP(I ) = 0 . 0 DEL2P(I)=0.0 CONTINUE READ CONSTANT AND C O E F F I C I E N T S R E A D ( 5 , 1 0 1 0 ) XNU READ(5,1015) RO.VEL.XLE  C DO 100 0 = 1.NN READ(5,1020) I,RC(I ) ,ZC(I ) WRITE(G.1020) I , R C ( I ) , Z C ( I ) CONTINUE  100 C C READ SYSTEM TOPOLOGY C DO 105 I = 1.NE READ(5,1025)J,(NODE(J,1$),I$=1,6) WRITEI6,1025)J,(NODE(U,1$),I$=1,6) 105 CONTINUE C C C READ NODE NUMBERS WITH S P E C I F I E D DISPLACEMENT C DO 110 1 = 1 .NNSRD READ(5,1030) NSRD(I) WRITE(6.1030) NSRD(I) 110 CONTINUE C C C READ TEMPERATURES C DO 120 J = 1.NN R E A D ( 5 , 1040) I , T ( I ) 120 CONTINUE C C PRINT DATA AND HEADLINES C WRITE(6.2010) WRITE(6.2020)RO.VEL.XLE WRITE(6,2030)XNU WRITE(6,2035)NN,NE,NNSRD C C C PRINT HEADLINES FOR DISPLACEMENTS C WRITE(G.2100) C C FORM MATRIX FOR I N I T I A L STRAIN AND S T I F F N E S S MATRIX C C C C C C C  GIVE VALUES TO WEIGHT FUNCTIONS AND NATURAL COORDINATES  FOR NUMERICAL  INTEGRARTION OF OUINTIC ALPHA 1=.0597 1587 178977 BETA 1 = . 4 7 0 1 4 2 0 6 4 105 1 15 ALPHA2=.797426985353087 BETA2 =.101286507323456  C W(1)=.225 W ( 2 ) = . 132 3 9 4 1 5 2 7 8 8 5 0 6 W(3)=W(2) W(4)=W(2) W(5)=. 125939180544827 W(6)=W(5) W(7)=W(5)  ORDER  414  XL( 1 , 1 )=1 . / 3 . XL( 1 ,2)=ALPHA1 XL( 1 ,3)=BETA1 XL( 1 , 4)=BETA1 XL( 1 ,5)=ALPHA2 X L ( 1 , G ) =BETA2 XL( 1 , 7 ) = B E T A 2 C XL(2. 1 ) = 1,/3. XL(2.2)=BETA1 XL(2,3)=ALPHA1 XL(2,4)=BETA1 XL(2,5)=BETA2 XL(2,6)=ALPHA2 XL(2,7)=BETA2 C  140 C C  DO 140 1=1,7 XL(3,I)=1.-(XL(1,I)+XL(2.I)) CONTINUE  DO GOO K=1,NE C c  180 C C  DO 180 I$=1 ,6 N(I$)=NODE(K,1$) RC$(I$)=RC(NODE(K,1$ ) ) ZC$(I$)=ZC(NODE(K,1$) ) CONTINUE  DO 185